On some older veneers, you can actually peel some outer layers off like peeling sections off a onion, but sometimes you cannot.
What I would think about would be to patch or float out the missing section or even cut the hole out square and apply a patch veneer in that area to have a smooth surface. float out any cracks.
Now you have a smooth sub surface, consider just adding a new layer of top surface. You can get a nice piece of finished out oak ply in 1/4 inch or even 1/8 inch. Contact cement this to the desk and use a router and smooth or round over the edges with a bit that has a roller.