What should a wife do with her husband who only prays on Fridays, while she desires a divorce but fears its consequences for their two children?

If the husband only prays on Fridays, then disliking him for this is correct because hating the perpetrators of sins is part of faith. Abandoning prayer is a great sin, and some scholars consider it disbelief (kufr). A woman has the right to seek divorce from a dissolute husband, but she can be patient with him, advise him, and supplicate for him. If he does not repent, then there is no good in remaining with him, and she must seek a divorce. If he refuses, she should refer the matter to a judge. Custody of the children belongs to the mother as long as she does not remarry, and if the father persists in his dissoluteness, he has no right to their custody.