My wife and I bought a "fruitwood" table several years ago. It has a fruitwood veneer to it that is starting to peel, and our kids like to do damage when they can It also has that spill catcher groove (not sure what the correct term is) that is annoying because of all the gunk that builds up in there. The table is pretty heavily built and so I want to refurbish is as opposed to replace it.

1. Any ideas on what we should use to fill in the groove before covering it?
2. Would buying a 1/4 inch sheet of some cabinet quality wood be better than using the veneer for the covering?

I would appreciate any advice to get us going in the right direction.