Welcome to the forums. I'm sorry your husband cheated on you, and while I commend you for looking within to discover ways you could have been a better partner, that in no way excuses your husband's affair. Secondly, this woman he is seeing seems to be dishonest and incapable of sticking to the rules. The very fact that she broke the non-communication rule while you and hubby were making a valiant attempt to repair your marriage shows me she has no respect for rules, and likely little real respect for your marriage. She comes across as a woman who will do anything and say anything to get what she wants.
I totally understand why the psychologist recommended getting her out of your life for a while. And while it's admirable you think this will make your husband sad, you're paying that pschologist for advice and insight, so perhaps you should listen to him/her and try the suggestion.
Good luck to you both.