Just, Before, no After yet

This kind of veneer peeling happens to sewing machine cabinets quite often.  It is not difficult to fix.  I like the pictured glue

There are many folks out there with a lot more experience and knowledge that I, this is just how I do it.

I first check, by pressing gently, that the veneer will push down into the proper place without further cracking.  If not, I put a damp rag on it while my mini iron heats up, then cautiously iron it flat.  I put a dry paper towel on top ot the ironed veneer, a flat board, and a heavy weight and let it cool under flattening forces.

Then I work some of the glue under the veneer with a nail file.  Carefully.  The I cover the area with waxed paper, ably the weight, and go away again.

the last picture is of a hinge that sadly broke  Hopefully, I'll find a replacement..