Do I need to fill the gap of a wide tv mount hole

Expanding foam can get messy but it may be your best bet, if you can keep it in place and they do not chew through it.

You may want to either incorporate a heavy gauge metal mesh wire, neatly cut around or into the place the mice go through. You might could even "bed" the wire mesh in the foam as it is expanding to lock in place that way. Doing this will make a potentially messy task and make it even messier.

This will not look nice in any sense of the word on the larger holes but it will certainly slow them down for a while. Otherwise a cat, as mentioned in the comments section, will also take care of the problem.

Thin sheet metal screwed in place will work as well without the foam too. You could try nailing it down, but in the cabinet it will bounce a bunch, screws are better here.

It sounds like you have solid 13/16" thick plywood for a wall. If that is the case, I'd drill holes matching holes in the mount all the way through, get four metal bolts, four matching nuts and eight washers, and hang the mount on the wall, securing with nuts from the opposite side of the wall and using threadlocker on bolt threads. You would have nuts permanently visible on the back side but the construction should be more than sturdy enough to hold a 40 lb TV.

If you don't have solid plywood but something else (I doubt that there's enough room for anything else but who knows?), that's a different story. You should find out as soon as you drill the first hole.