What is the ruling on staying with a mother who married a Christian who hates Islam and the Prophet, and who has abandoned prayer? Is it advisable to stay with her or to leave the home?

The marriage of a Muslim woman to a non-Muslim is invalid and forbidden by Sharia (definition), and her cohabitation with him is considered fornication. It is not obligatory to remain in a house where there is an abomination that cannot be changed, such as the existence of this forbidden relationship; rather, it is better to leave it. Nevertheless, one must continue to be dutiful to one's mother, treat her kindly, and advise her, without obeying her in disobedience to Allah.