If you had it written for him there are things you can do to show cohabitation such as social media and time spent there.  It could require a private investigator and can be expensive.

Best you can do is create the best place for your kids when they are with you. Make them know your house is consistent, safe and they come first.   I always try to not listen to anything about the other house less I get upset.  Your kids don't have to like the partner only tolerate him.

Hello HRE, 
I am sorry you are going through this.  

Based on my own experience and that of people I know, you will probably have no success at all if you go to court.  Family court judges are not concerned about personal lifestyles or any living arrangements you and your ex agreed to.  If your children are physically in danger - that's another story - but that doesn't sound like the case here. 

In my opinion, and, it's only an opinion, the best thing you can do is set an example for your children.  If you don't approve of co-habitation, don't do it yourself.  Always be there for your kids.  Don't bad-mouth the dad or his lover.  Children are smart and if your ex is a jerk, they will see that for themselves.  Be kind to others and if you're not religious, have some spiritually.  Hold your head high and tell yourself that this will pass.  Because it will.  One day your children will be grown and they won't remember what was said or done, but they will remember how the situation made them feel.  

I know it's hard but I think you will feel better in the long run if you try to not be angry.  Again, this is only my opinion. 

I wish you the best during this trying time.