File /boot/vmlinuz-5.10-x86_64 not found

[manjaro-gnome /]# sudo pacman -Syu $(pacman -Qq | grep -i "^linux[0-9]")

:: Synchronizing package databases...
 core is up to date
 extra is up to date
 community is up to date
 multilib is up to date
error: target not found: linux510
error: target not found: linux510-headers
warning: linux510-virtualbox-host-modules-6.1.30-4 is up to date -- reinstalling
error: target not found: linux54
error: target not found: linux54-headers
warning: linux54-virtualbox-host-modules-6.1.30-4 is up to date -- reinstalling

seems like it is not finding the headers

Of course!

[manjaro-gnome /]# mkinitcpio -P
==> ERROR: No presets found in /etc/mkinitcpio.d
[manjaro-gnome /]# mhwd-kernel -li
Currently running: 5.9.16-1-MANJARO (linux59)
The following kernels are installed in your system:
   * linux510
   * linux54
[manjaro-gnome /]# mhwd-kernel -l
available kernels:
   * linux510-rt
   * linux515-rt

Still in chroot try:

sudo pacman -S linux54 linux54-headers linux510 linux510-headers
mkinitcpio -P 

then sudo update-grub and reboot

can you just add kernel 5.15
you should have sames number version kernels in /boot/ , /modules and *.preset

by chroot ( boot on USB iso live manjaro , open a terminal and a browser on this topic

sudo manjaro-chroot -a
pacman -Syy ( check that inter is ok ) 
pacman -S linux515 linux515-headers
mhwd -li 
ls /etc/mkinitcpio.d/*.preset
ls /boot/
ls /usr/bin/modules/
mkinitcpio -P
exit ( to end chroot )
[manjaro-gnome /]# pacman -S linux54 linux54-headers 
error: target not found: linux54
error: target not found: linux54-headers

It sadly does not find them

[manjaro-gnome /]# pacman -Syy
:: Synchronizing package databases...
 core                               137.7 KiB   678 KiB/s 00:00 [##################################] 100%
 extra                             1918.4 KiB  4.46 MiB/s 00:00 [##################################] 100%
 community                            6.8 MiB  9.53 MiB/s 00:01 [##################################] 100%
 multilib                           176.5 KiB  8.62 MiB/s 00:00 [##################################] 100%
[manjaro-gnome /]# pacman -S linux515 linux515-headers
error: target not found: linux515
error: target not found: linux515-headers

Same error here.

can you report commands

mhwd -li 
ls /etc/mkinitcpio.d/*.preset
ls /boot/
ls /usr/bin/modules/
[manjaro-gnome /]# mhwd -li
> Installed PCI configs:
--------------------------------------------------------------------------------
                  NAME               VERSION          FREEDRIVER           TYPE
--------------------------------------------------------------------------------
           video-linux            2018.05.04                true            PCI


Warning: No installed USB configs!
[manjaro-gnome /]# ls /etc/mkinitcpio.d/*.preset
ls: cannot access '/etc/mkinitcpio.d/*.preset': No such file or directory
[manjaro-gnome /]# ls /boot/
 EFI  'System Volume Information'
[manjaro-gnome /]# ls /usr/bin/modules
ls: cannot access '/usr/bin/modules': No such file or directory
[manjaro-gnome /]# 

under chroot

[manjaro-gnome /]# mhwd-kernel -li
Currently running: 5.9.16-1-MANJARO (linux59)
The following kernels are installed in your system:
   * linux510
   * linux54
[manjaro-gnome /]# ls /boot/*
/boot/EFI:
Manjaro  boot

'/boot/System Volume Information':
[manjaro-gnome /]# ls /lib/modules/*
/lib/modules/3.16.81-1-MANJARO:
updates

/lib/modules/3.16.82-2-MANJARO:
kernel  modules.builtin.alias.bin

/lib/modules/5.10.79-1-MANJARO:
build         modules.alias              modules.builtin.bin      modules.devname  modules.symbols.bin
extramodules  modules.alias.bin          modules.builtin.modinfo  modules.order    pkgbase
kernel        modules.builtin            modules.dep              modules.softdep  updates
kernelbase    modules.builtin.alias.bin  modules.dep.bin          modules.symbols  vmlinuz

/lib/modules/5.4.144-1-MANJARO:
kernel  updates

/lib/modules/5.4.159-1-MANJARO:
build         modules.alias              modules.builtin.bin      modules.devname  modules.symbols.bin
extramodules  modules.alias.bin          modules.builtin.modinfo  modules.order    pkgbase
kernel        modules.builtin            modules.dep              modules.softdep  updates
kernelbase    modules.builtin.alias.bin  modules.dep.bin          modules.symbols  vmlinuz

/lib/modules/extramodules-5.10-MANJARO:
vboxdrv.ko.xz  vboxnetadp.ko.xz  vboxnetflt.ko.xz  version

/lib/modules/extramodules-5.4-MANJARO:
vboxdrv.ko.xz  vboxnetadp.ko.xz  vboxnetflt.ko.xz  version

so pacman -S linux515 do not install ?

Ǹo, it does not find the target…

so you will create thes files

nano  /etc/mkinitcpio.d/linux510.preset 
# mkinitcpio preset file for the 'linux510' package

ALL_config="/etc/mkinitcpio.conf"
ALL_kver="/boot/vmlinuz-5.10-x86_64"

PRESETS=('default' 'fallback')

#default_config="/etc/mkinitcpio.conf"
default_image="/boot/initramfs-5.10-x86_64.img"
#default_options=""

#fallback_config="/etc/mkinitcpio.conf"
fallback_image="/boot/initramfs-5.10-x86_64-fallback.img"
fallback_options="-S autodetect"
nano  /etc/mkinitcpio.d/linux54.preset 
# mkinitcpio preset file for the 'linux54' package

ALL_config="/etc/mkinitcpio.conf"
ALL_kver="/boot/vmlinuz-5.4-x86_64"

PRESETS=('default' 'fallback')

#default_config="/etc/mkinitcpio.conf"
default_image="/boot/initramfs-5.4-x86_64.img"
#default_options=""

#fallback_config="/etc/mkinitcpio.conf"
fallback_image="/boot/initramfs-5.4-x86_64-fallback.img"
fallback_options="-S autodetect"

then

mkinitcpio -P

Found the exact files as *.preset.pacsave files and copied them to the correct name.

Result:

[manjaro-gnome /]# mkinitcpio -P
==> Building image from preset: /etc/mkinitcpio.d/linux510.preset: 'default'
  -> -k /boot/vmlinuz-5.10-x86_64 -c /etc/mkinitcpio.conf -g /boot/initramfs-5.10-x86_64.img
==> ERROR: specified kernel image does not exist: `/boot/vmlinuz-5.10-x86_64'
==> Building image from preset: /etc/mkinitcpio.d/linux510.preset: 'fallback'
  -> -k /boot/vmlinuz-5.10-x86_64 -c /etc/mkinitcpio.conf -g /boot/initramfs-5.10-x86_64-fallback.img -S autodetect
==> ERROR: specified kernel image does not exist: `/boot/vmlinuz-5.10-x86_64'
==> Building image from preset: /etc/mkinitcpio.d/linux54.preset: 'default'
  -> -k /boot/vmlinuz-5.4-x86_64 -c /etc/mkinitcpio.conf -g /boot/initramfs-5.4-x86_64.img
==> ERROR: specified kernel image does not exist: `/boot/vmlinuz-5.4-x86_64'
==> Building image from preset: /etc/mkinitcpio.d/linux54.preset: 'fallback'
  -> -k /boot/vmlinuz-5.4-x86_64 -c /etc/mkinitcpio.conf -g /boot/initramfs-5.4-x86_64-fallback.img -S autodetect
==> ERROR: specified kernel image does not exist: `/boot/vmlinuz-5.4-x86_64'

if you try
pacman -Syyu

[manjaro-gnome /]# pacman -Syyu
:: Synchronizing package databases...
 core                               137.7 KiB   668 KiB/s 00:00 [##################################] 100%
 extra                             1918.4 KiB  2.28 MiB/s 00:01 [##################################] 100%
 community                            6.8 MiB  5.22 MiB/s 00:01 [##################################] 100%
 multilib                           176.5 KiB  5.07 MiB/s 00:00 [##################################] 100%
:: Starting full system upgrade...
resolving dependencies...
looking for conflicting packages...

Packages (1) pacui-1.14.r39.ga37e948-1

Total Download Size:   0.06 MiB
Total Installed Size:  0.21 MiB
Net Upgrade Size:      0.00 MiB

:: Proceed with installation? [Y/n] Y
:: Retrieving packages...
 pacui-1.14.r39.ga37e948-1-any       65.8 KiB   766 KiB/s 00:00 [##################################] 100%
(1/1) checking keys in keyring                                  [##################################] 100%
(1/1) checking package integrity                                [##################################] 100%
(1/1) loading package files                                     [##################################] 100%
(1/1) checking for file conflicts                               [##################################] 100%
(1/1) checking available disk space                             [##################################] 100%
:: Processing package changes...
(1/1) upgrading pacui                                           [##################################] 100%
New optional dependencies for pacui
    pamac-cli: One AUR helper is needed for AUR support [installed]
    yay: One AUR helper is needed for AUR support [installed]
:: Running post-transaction hooks...
(1/1) Arming ConditionNeedsUpdate..```

This is the result from it. Maybe the update itself was just unlucky but otherwise that happened

then check number carefully

pacman -Ss linux510

then try ( n to use see line before )

pacman -U linux510-headers-5.10.n-x86_64.pkg.tar.zst
pacman -U linux510-5.10.n-x86_64.pkg.tar.zst
[manjaro-gnome /]# pacman -U linux510-headers-5.10.n-x86_64.pkg.tar.zst
loading packages...
error: 'linux510-headers-5.10.n-x86_64.pkg.tar.zst': could not find or read package

Got a lot of output for the first one though, but no headers found in the list

what you get for
pacman -Ss linux510 ?

[manjaro-gnome /]# pacman -Ss linux510
extra/linux510-acpi_call 1.1.0-100 (linux510-extramodules)
    A linux kernel module that enables calls to ACPI methods through /proc/acpi/call
extra/linux510-bbswitch 0.8-100 (linux510-extramodules)
    kernel module allowing to switch dedicated graphics card on Optimus laptops
extra/linux510-broadcom-wl 6.30.223.271-102 (linux510-extramodules)
    Broadcom 802.11 Linux STA wireless driver BCM43142.
extra/linux510-nvidia 495.44-10 (linux510-extramodules)
    NVIDIA drivers for linux.
extra/linux510-nvidia-390xx 390.144-33 (linux510-extramodules)
    NVIDIA drivers for linux.
extra/linux510-nvidia-470xx 470.86-7 (linux510-extramodules)
    NVIDIA drivers for linux.
extra/linux510-r8168 8.049.02-37 (linux510-extramodules)
    A kernel module for Realtek 8168 network cards
extra/linux510-rtl8723bu 20211104-6 (linux510-extramodules)
    A kernel module for Realtek 8723bu network cards
extra/linux510-tp_smapi 0.43-100 (linux510-extramodules)
    Modules for ThinkPad's SMAPI functionality
extra/linux510-vhba-module 20211023-8 (linux510-extramodules)
    Kernel module that emulates SCSI devices
extra/linux510-virtualbox-host-modules 6.1.30-4 (linux510-extramodules) [installed]
    Host kernel modules for VirtualBox
extra/linux510-zfs 2.1.1-18 (linux510-extramodules)
    Kernel modules for the Zettabyte File System.
community/linux510-rt 5.10.83_rt58-1
    The Linux510-rt kernel and modules
community/linux510-rt-acpi_call 1.1.0-65 (linux510-rt-extramodules)
    A linux kernel module that enables calls to ACPI methods through /proc/acpi/call
community/linux510-rt-bbswitch 0.8-8 (linux510-rt-extramodules)
    kernel module allowing to switch dedicated graphics card on Optimus laptops
community/linux510-rt-broadcom-wl 6.30.223.271-9 (linux510-rt-extramodules)
    Broadcom 802.11 Linux STA wireless driver BCM43142.
community/linux510-rt-headers 5.10.83_rt58-1
    Header files and scripts for building modules for Linux510-rt kernel
community/linux510-rt-nvidia 495.44-4 (linux510-rt-extramodules)
    NVIDIA drivers for linux.
community/linux510-rt-nvidia-470xx 470.86-3 (linux510-rt-extramodules)
    NVIDIA drivers for linux.
community/linux510-rt-r8168 8.049.02-6 (linux510-rt-extramodules)
    A kernel module for Realtek 8168 network cards
community/linux510-rt-rtl8723bu 20211104-3 (linux510-rt-extramodules)
    A kernel module for Realtek 8723bu network cards
community/linux510-rt-tp_smapi 0.43-8 (linux510-rt-extramodules)
    Modules for ThinkPad's SMAPI functionality
community/linux510-rt-vhba-module 20211023-4 (linux510-rt-extramodules)
    Kernel module that emulates SCSI devices
community/linux510-rt-virtualbox-host-modules 6.1.30-2 (linux510-rt-extramodules)
    Host kernel modules for VirtualBox

Just cross-checked with my running instance on another device. Seems like core is missing right?
In more detail these lines:

core/linux510 5.10.84-1 [installed]
    The Linux510 kernel and modules
core/linux510-headers 5.10.84-1
    Header files and scripts for building modules for Linux510 kernel