Veneer Desk Top

Here are some ways to fix peeling veneer on a desk:
Bondo
Use a putty knife to apply a small amount of bondo putty and a pea-sized amount of hardener, then stir to combine. Apply more putty than you think you'll need, and sand it smooth when it hardens. You can use lacquer thinner to clean off dust, and then apply primer.
Contact cement
Apply contact cement to both bonding surfaces, and let it dry. Then, place the pieces together, smooth it out with a J roller, and trim the edges with a router. Finally, sand the surface. Contact cement works well for a short dry time, but it doesn't allow for flexibility.
Glue
Use a wood glue like Elmer's, Titebond, or Gorilla Wood Glue. You can use a toothpick to spread the glue into nooks and crannies, and then press down with your fingers. Wipe away excess glue, and then clamp or weight down until the glue is dry.
Paint stripper
Use paint stripper to remove the defective finish, and then clean with mineral spirits.