This part sounds straightforward enough though: "I want to see if I can get a provision in today's motion to have us both attend separate counseling with a professional on how to deal with divorce and affairs with the children, beyond the required class. The psychiatrist that I saw is part of a group--she said she would have someone see him, and they can confer about the case."
Talk to the mediator, not H. Get rid of the blaming language.
"I would like a provision for us both to attend separate counseling with a professional on how to deal with divorce and new relationships with the children. My therapist recommends that (s)he confer with someone who has met with H about the case, so that they can figure out the best approach for the kids."