How can I stain a white spot on wood?

Could you replace the veneer that was there with a new veneer?

You might try roughing up the white spot with coarse sandpaper and then using a gel stain. That works on some pieces but not all. I read somewhere that you can cover the spot with a thin coat if shellac and then again over it. I would try it on a scrap piece of wood first.

I had this same problem on several doors in the house we're refurbished for my daughter's college years. Although sanding, and more sanding, certain areas would not take stain. I ended up matching paint as closely to the door colors throughout the rest of the house and painted the odd doors and had to use stain to add faux graining. It wasn't easy, but no one can tell it isn't real grain unless I tell them.

You can use gel stain. It's thicker than regular stain. Any good gel stain will work. AI like Minwax products

https://www.minwax.com/how-to-finish-wood/change-stain-color-with-polyshades/

https://www.minwax.com/woodworking-videos/application-how-to/one-step-wood-stain-finish-minwax-polyshades

https://www.youtube.com/watch?v=e2JG_l9WScM

Mayonnaise Apply a dab with a soft cloth and rub it into the mark in a circular motion. ...

Thanks