I understand that Cohanim are prohibited from marrying a woman who has had prior relations with a non-Jew, even if she did not marry him. However, I was also told that there are "exceptions" that can be made to this, even though I’ve never heard of such exceptions being made. I was wondering what scenarios might constitute an exception?

I don't know of an exception. The following is found in Shulchan Aruch EH 6, 12-13; maybe that is what you are looking for. If a Cohen marries his wife that had premarital relations with a non-Jew, and she or a witness reveilles this information to him only after the wedding, since there are no two valid witnesses, he doesn't have to believe her and does not need to divorce her.