Personally, I would let your children hang out with ow on the condition that they break things in her home. For example, if they broke a lamp, they would get to stay up late. If they broke ow's favorite lamp, then they get a new toy. I know you are tired. However, I have trouble with the ow having time with YOUR kids. To me, it legitimizes what your h is doing because everyone is happy and everyone accepts what is going on.
If there is a divorce and a divorce settlement, then you have no choice. Right now, you and your h are married. It doesn't say that in the case of adultery, you get to share the kids with ow. In my case, I don't think my kids (one is 11 and the other is 15) are ready to start a relationship with a new father. In fact the oldest has already told me that in the case of a divorce she doesn't want to live with her mother because she is afraid of the men she would invite to the house without me around to protect her.
My younger daughter may go along because she is still at the age where she wants to please. That phase is about to change. In fact, in about a year from now, I could easily end up with full physical custody of the children. Then she would have visitation at best.
As I stated before, I know you are tired. I posted before that you need a break. But if your kids start hanging out with ow, are you ready to hear about her from your kids? Are you ready to hear how nice she is? That is something you have to prepare yourself for and it will hurt.
I am not an expert on this, but I feel you are frustrated and tired, but you really don't want ow to be involved with your kids. (((Hugs)))