What is the ruling regarding a wife remaining with her husband who is negligent in performing prayers, given her fear for her children being influenced by him, and her inability to seek a divorce due to the lack of a source of income or shelter?

Prayer is the pillar of Islam. Whoever abandons it out of denial has committed an act of disbelief by scholarly consensus. As for one who abandons it out of laziness while acknowledging its obligation, the majority opinion is that he does not exit the fold of Islam. Since your husband does not deny the obligation of prayer, he is a Muslim, and it is permissible for you to remain with him. You should strive to reform him with gentleness and wisdom, reminding him of Allah and the Last Day, seeking help from righteous people, and increasing your supplications for him.