[Arran]

That cabinet damage is pretty severe, that was probably why the set was bargain priced, the previous owner just didn't want to deal with it. My guess is that the laminations of veneer were already loose from moisture damage and something snagged that back corner and the veneer was ripped off and lost.
The good news is that I think it is repairable, but the first step will be to remove the top layer of mahogany (or whatever species it actually is), and patch and glue the layers underneath. One thing I will suggest is that you invest in a large collection of bar clamps and a larger hypodermic needle and syringe for glue injection. The wood underneath was likely a type of plywood composed of layers of a cheap hardwood veneer like birch, beech or maple, it is usually fairly inexpensive and easy to find from various sources including fleabay.
The only close substitute for this hardwood plywood is Baltic birch plywood, but unless the side is totally disintegrating and falling off I think repairing what's there, as opposed to trying to make a replacement, would be the best solution. Taking a cabinet apart that isn't already coming apart is not something you want to try unless you've had a lot of experience with cabinet repair. As for trying to reproduce cabinet parts I would say the same thing, it can be very infuriating trying to mate a new panel into an old cabinet sometimes.
Regards
Arran