I have had this happen to me a few times. Tiny marks can sometimes be dabbed out, gently, with a moist(ish) tissue. You could also try scraping ever so gently with a very sharp craft blade. I use the blade alone and not held in the knife handle as that gives you a finer, softer control over what you are doing. Part of the problem is that any degree of effort to take away the mark alters the surface of the mount card and that will show even though the stain may have gone.
As I cut my own mounts and have stocks of mount card I usually cut a new mount if the stain/mark is at all larger. You may have to make your mind up to have a new mount, then bravely attempt to remove the mark and maybe, succeed! Good luck.