Repairing veneer that has never been repaired is certainly easier
than repairing veneer that someone has already tried to repair that has
failed, or you would not need to do it again.

If you are repairing veneer on a piano that has never been repaired,
then look closely at the glue that was originally used.  It should
scrape off easily into a fine powder that indicates hot hide glue.

Hot hide glue is always my choice for veneer when redoing a fine old
original antique as this was the original glue.  It fails due to
several factors such as old age or in some cases there just wasn't
enough glue in the beginning to hold the veneer forever or for some
other unknown reason.

If you listen to your woodworking friends they will suggest contact
cement.  It is easy to work with and will be a temporary repair no
matter what anyone tells you.  If you try woodworkers glue such as the
yellowish glue sold in stores then it will make a gummy mess as it does
not bond well with hide glue.

I would clean the area and use hot hide glue.  You can clamp the veneer
with heavy felt padding between the veneer and the clamp.  Light pressure
is all that is needed.  Let it dry overnight and the next day you can
polish off any excess glue with polish.

Don Teach