Repair a Hole/Crack in Hollow Core Door
You can repair it but you're not going to be able to match the wood decor grain - which means you'll have to paint it.
There are a few different products you can use but I like Bondo. It's basically the same stuff they use for auto body repair. It looks like there is enough backing in the hole that you shouldn't need anything but the Bondo, a putty knife and a few sheets of sandpaper. Apply it according to directions trying not to overfill it too much since you'll want to sand it smooth with the door. After it dries start with about a 120 grit sandpaper or sanding block to take off most of the excess and finish with a 220 or higher grit to finish. If there are any hollows in the patch you can apply more Bondo after it's dry. It will be difficult to get the patch to totally disappear but with patience and going over it a few times you can get it very close.
It's hard to tell the material in the picture but you may need a good prime coat to get the adhesion and consistency you'll want with the paint.