What is the ruling on supplicating for the mother's divorce so she may return to the father, and how can atonement be made for the sin of shaming a girl and causing the ruin of a household?

The basic principle is to conceal a Muslim's faults and not expose them, as the Prophet, peace and blessings be upon him, said: "Whoever conceals a Muslim, Allah will conceal him on the Day of Resurrection." If the Muslim is of concealed حال (state/condition) and not known for sins, then it is obligatory to conceal him and unlawful to expose him. Repentance to Allah is obligatory, and seeking forgiveness from the father and the girl is also obligatory, unless a greater harm would result from it. One should also increase in supplication and seeking forgiveness for them. As for supplicating for the mother to be divorced so she can return to the father, that is an unlawful supplication.