I would believe the best course of action would be to sand with the grain using Flexovit 100mm x 1m 400 Grit Wet And Dry Sandpaper Roll. I'm not sure of what the coating is but in order for you to repair just this section it would need to be similar to what has been used previously. It is quite possibly a Shellac coating. Alternatively, you could sand the whole top back with a Sandblaster™ 320 Grit Fine Between Coats Sanding Sponge. You'll then be able to coat with an interior varnish.
Please let me know if you need further information or had questions.
In order to see if this has a Shellac finish or a varnish grab some metho on a cloth and give a firm rub. If its shellac it will become sticky if its varnish or a poly type product it will have a white finish to it. If it shellac you will need to do some research on how to restore a shellac finish or sand back and use a poly product
I suppose it depends on what sort of end result you're wanting. It's an interesting looking desk that could benefit from a complete restoration. Or for a quick fix you could just sand back the border edge trims and recoat with a varnish.