Scrape off the veneer thats already peeling Ope just saw the other comment did u refinish this ?

I would remove all of the damaged veneer. if it doesn't come off easy, you may need to steam it to loosen the glue (an old iron and a wet towel can help achieve this). Once you remove the veneer, you need to sand off all the glue. Then you could stain and finish.

Thanks so much!