Clay, Some questions about where the holes are located.
Is this for a CEO type office and the holes are on the front (as on the part everyone sees) of the desk where everyone who walks in might notice a repair?
Or, are the holes on the sides or back where it’s likely nobody would notice...and I mean nobody would notice (besides yourself) even if you didn’t repair the holes.
I’m no expert (I’ve done a couple) but under the majority of circumstances, Charlie’s suggestion will be by far the easiest and probably 100% effective.
OTOH, a reasonably done veneer patch also can work, but you are increasing the number variables that can end up requiring attention. For example, grain, surface texture, color, top finish and general blending of the area.
Personally, I would start with the crayons (and this part is important) and then live with that repair for quite awhile before I would move towards a patch. In my experience, what seems obvious at first glance, will totally disappear the second you quit focusing on the repair.
Assuming you don’t have severe OCD!!!
Besides, you can try the crayons first, and still do a patch if you hate the crayons.