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.