You're obviously upset about the whole "paramour" business. It might be a good idea to talk through your hurt, betrayal, disgust, etc. with a counselor--therapist, religious advisor, trusted friend, etc.--to deal with your emotional issues.
But better to leave all of that out of the divorce and child custody issue. Even though you're disgusted with your wife's behavior and hurt by the betrayal, it's better to treat the divorce like a business deal and try to get it done with quickly, efficiently and fairly. Trying to "punish" her for her moral failures will just make getting divorced harder, more unpleasant and probably more expensive.
You might point out to your wife that you think her behavior is likely to be confusing to the kids, especially if they don't yet know about the divorce, and ask her politely to cool it when they're around. For that matter it wouldn't be out of line to politely ask her to cool it when you're around, just out of respect for your feelings. What she does in private is her business, you're better off putting that out of your mind as much as you can.
If you think she's putting your kids in a bad environment with the other guy, then discuss with your lawyer about what your rights and options are.