That was my mistake, my apologies. You were right in adding -kernel to the command.
But let’s try updating the database first. From within the chroot environment, do:
pacman -Syyyu
Along with any software from the repositories, this should update the package database, which is what we’re after. Then try
sudo mhwd-kernel -li
again, to see what kernels are installed and also try:
sudo mhwd-kernel -l
to get a list of available kernels.
(You’ll probably have notice and wondered about the sudo prefix, seeing as we’re already root, but my opinion is “rather safe than sorry” and it hasn’t let me down yet.)
You can then try installing kernel 5.10 again, as previously advised:
sudo mhwd-kernel -i linux510
and go from there.
Also see How to remove a "ghost-kernel"?
Hope this helps! If not, I’m fresh out of ideas and won’t be long here anyway, before I’ve got to be off.