That spot could very well be a wound from bumping into a sharp imperfection in a tank decoration. It is in the prime position on his body to be an injury like that, on the top of his head. That exact location is where I have seen most betta injuries, with the exception of the fins. Remove the decorations from the tank and test them to see if they are betta safe by running some nylon panty hose across the rough edges. If it snags the pantyhose, it's not safe for your betta, and needs to be filed down or sandpapered if possible. Plastic pointy plants are notorious for this, and are best avoided. Also run your finger along the inside of plastic caves, and along the seams of plastic decorations, and look for sharp imperfections. You might be able to remove the imperfection without junking the entire decoration.
There's a couple of problems I'm seeing with introducing a cory to this tank. 5 gallons is a good size for a betta with NO TANK MATES, but it's just not enough space to add tank mates as bettas are quite territorial. You *might* be able to get away with a snail or a shrimp, but that'd be about it. And that's not likely considering the betta already lives there, so no matter what you introduce to the tank, it will be an intruder invading his territory. Also, corys are a shoaling fish that need to be in groups of at least 5 or 6 minimum. So a 5 gallon tank is too small for a shoal of corys on their own, even without a betta. If you are concerned that your betta is lonely (he's not) and you want to get him a friend, I would look at adding a Nerite Snail, as they don't reproduce in fresh water, so you won't get an infestation. But I would remove the betta from the tank and rearrange it (so it's like a different territory), then introduce the snail, then re-introduce the betta.