If the piece of veneer is loose and lifting slightly you just need to make sure it is clean on the underside by sliding a thin knife or spatula. Blow some air behind it if possible to remove any dirt. Then using the same knife or spatula apply some wood glue or Elmers white glue (unless you have hide glue) to the back side and use a clamp or even a tightly wrapped piece of tape to hold the veneer flat. I recommend that you first try to “dry” fit the veneer into position before applying the glue to make sure the fit is as good as possible.
Remember to have a professional do the job correctly if you are not up to doing it yourself, especially when you have a valuable antique or piece of fine furniture. We’ll come and give you a free consultation.