Is it permissible to secretly marry a second wife when the husband needs it and fears falling into sin, and the first wife threatens to demand divorce, abandon the children, and make their lives miserable if polygyny is mentioned?

It is permissible for a man to marry a second wife even if his first wife is unaware, provided he is capable of being just and providing for both. It may even be obligatory for him if he fears falling into sin. He should advise his wife gently and kindly, and inform her that marrying a second wife is something Allah has permitted.