[Music in a bottle]

I'm tempted to gently remove the peeling veneer, to refinish the wood below. Good idea or no?

Or no.

The wood underneath the (usually walnut) veneer is a cheap white wood that, in general, has little beauty and is only good for structure, not for appearance.

So, What would you recommend as a treatment for that area? I can attempt to put up some pictures of the affected area. Doing a single curved radius would be easily doable, but a dual curved type radius, well, I sense this will require more than some scissors and paste! I'm already looking for the wife's bottle of Jim Beam! We happen to really like this particular radio console.