What is the ruling on someone who was forced to falsely swear upon the Mushaf, fearing the ruin of his home and the divorce of his wife, knowing that they have children?

Lying is forbidden and a major sin. Its sinfulness increases if accompanied by an oath and placing one's hand on the Qur'an. Resorting to outright lying is not permissible except out of necessity or a need that cannot be met through another lawful means, and circumlocutions (ma'areedh) suffice to avoid lying. It appears there was no justification for the husband's lying and oath in this instance, and he could have used circumlocutions. Therefore, he must repent. It is not permissible for the wife to object to her husband marrying another, nor to seek divorce on that account, because Allah has permitted a man to marry up to four.