Should a woman seek divorce and cause her two daughters to become displaced, or should she patiently endure her husband's betrayal, as he is secretly involved with another woman?

You must advise your husband and warn him of Allah and His punishment if you are certain of his illicit relationships. You can also suggest he marry another woman to keep himself chaste, as this is better than divorce. First, you must verify the illicit nature of the relationship. It is not permissible to seek a divorce unless you are certain of his persistence in sin, for the Prophet, peace and blessings be upon him, said: "Any woman who asks her husband for divorce without any sound reason, the scent of Paradise will be forbidden to her." Patience with him, accompanied by advice, is more deserving, even if his involvement in such acts is proven.