You need to start by sanding. Be very careful not to sand through the veneer. Is it a polyurethane coat or similar "plastic" finish? If so, only sand through that. I'd start with 120 grit with a random orbit sander. Wear necessary safety gear of course including a particle mask.
You can try only sanding a feathered area around the damage. To match the finish, try sanding off some finish on the bottom or less visible area and applying a few stains. When these are completely set, sand them back off and reapply the one that matched best. An inside surface of the solid wood might be best to reduce risk of sanding through the veneer.
Or, just sand it smooth and paint it.