Is the wife entitled to seek khulʿ or divorce despite the husband's eagerness not to neglect her, and is the husband compelled to do so by Islamic law, and what is her status after leaving the house without the husband's knowledge?

Marrying a second, third, or fourth wife is permissible in Islamic law, as long as the husband is capable of upholding justice. Polygyny is permissible without a specific reason, so how much more so when there is a reason, such as increasing progeny. The wife must fear Allah and should not seek divorce without a compelling reason. If she dislikes her husband, she may seek khul‘ (dissolution of marriage). Otherwise, she is threatened with a severe warning. A wife leaving her home without her husband's permission is a grave transgression, and she is not entitled to stipulate receiving anything in return for her return. Therefore, she must repent to Allah and return to what is right. As for the decision to marry another woman in the hope of having a child, it is a sound one.