*I don’t know what the male equivalent of “side-chick” is for a guy

I know that I’m not the asshole for not knowing, and I was rather surprised and angry for her revealing it to me the other day. She had always had been a truthful person who would tell me everything about her, but she claims that she “never had the chance to tell me at an appropriate time” which granted, I’m rather busy most of the time and the activities we engage in aren’t particularly the most productive time to bring such a topic up. However, she and I agree that she had broken our trust and had to rethink this whole relationship thing (in other words, probably split), as she still has feelings for that boy she’s been dating for far longer than I have.

However, I’m torn. Truthfully, I really like this girl and she had helped me go through some rough times when I had trouble with my mental health. She also makes mistakes all the time. This one particularly affects me in a negative way but I’m verging on considering forgiving her for it, as I really liked where my relationship with her was going, and am unfortunately unwilling to give it up. 

She seems to agree, but she cannot bring herself to dump the third party. I can’t understand what’s going through her head right now, but assuming that she does not break up with the guy, and I continue this relationship, aware of the fact that I’m the “side-boy*” for her, would I be the asshole in this situation?

Tl;dr Am I the asshole for wanting to continue dating a girl that I had just learned had been cheating on another guy- with me, making me the male equivalent of a side-chick?