One think I like about polyamory is that relationships can be more fluid. You can go from a sexual relationship to friends without benefits if that is the direction it is heading. But it is all about talking about what you want.
Your husband's relationship with your gilrfriend should roughly be independent of your relationship with him. It sounds like you are feeling insecure. Him breaking up with her won't fix that problem, it will just make it easier to ignore. Talk to him about what he likes about you. See why he wants to be with you.
In theory, his relationship with that other woman should not affect his relationship with you. Once you start pushing for a choice, then it will affect both relationships.
It sounds like the root of the problem is communication. Try to work with him to communicate. A marriage counselor can be good at helping here.