Is it permissible to supplicate with the words: "O Allah, grant my husband contentment with women, and come between him and internet girls," for one who seeks marriage through unlawful means and speaks with women via the internet and phone, and dates them, despite his shortcomings in observing religious commitment, even though his wives fulfill his rights and he acknowledges that?

There is no harm in supplicating with the mentioned supplication. The husband should be gently advised that conversing with and dating non-mahram (definition) women is impermissible, and he must fear Allah and adhere to the bounds of Sharia (definition). If he is in need of marriage, he should pursue legitimate channels and avoid situations of suspicion. You, for your part, should strive to be a good wife to him; this will help him accept the advice.