Is it permissible for a wife to seek forgiveness for her husband who does not pray or fast, and for whom she hopes guidance, and will Allah forgive his sins through this seeking of forgiveness?

Not praying and not fasting is a grave sin, greater than adultery, theft, drinking alcohol, and killing a soul. Some scholars even consider it to be an act that expels one from the fold of Islam. The wife should advise her husband to repent and become righteous. If he responds, then that is good; otherwise, there is no good in remaining with him. Seeking forgiveness and supplicating for his guidance is permissible as long as he is considered a Muslim, because the Prophet, peace and blessings be upon him, commanded seeking forgiveness for believing men and believing women, and seeking forgiveness is only forbidden for polytheists. The ignorant person who abandons prayer is considered a Muslim by the majority of scholars, so it is permissible to seek forgiveness for him and supplicate for his guidance, and this matter is left to Allah Almighty.