I have had to perform wooden puzzle repair a few times in recent months. This sometimes becomes necessary with puzzles that are made by layering thin pieces of wood together. Inevitably, the top layer peels a bit away when little fingers are taking the tight pieces apart.
Dab a bit of your favorite crazy glue in between the wood parts that are peeling away. Clamp the puzzle piece with one of these nifty binder clips.
Let it dry thoroughly, and let the games begin!