This reveals the suspected - you are already in an out-of-date (or worse: partial-upgrade) state.
You need to fix that situation first .
(and we should always be updated/synced before installing new packages)
sudo pacman-mirrors -g && sudo pacman -Syyu
sudo mhwd-kernel -i linux510
Then reboot into 5.10 (tap Esc during boot to show grub)
Double check your running kernel is not 5.13 then remove it:
uname -r
sudo mhwd-kernel -r linux513
Then I would guess you can do everything else you want.