What is the ruling of the Sharia concerning a Muslim man who is married to a non-Muslim woman and has a daughter with her who lives in her mother's country, and is it permissible for him to kill the child for fear of her going astray in her mother's country?

First, the idea of killing the daughter must be abandoned, due to the grave sin involved. Second, this problem underscores the danger of residing in lands of disbelief and marrying women from the People of the Book. Third, the wife must reside where her husband resides, and she does not have the right to refuse to move or prevent him from taking the children. If the wife does not comply, divorce should be considered, but only after luring her to a Muslim country to obtain custody of the daughter, because disbelief nullifies the mother's right to custody. If that is not possible, then efforts must be made to take the daughter from her mother by any possible means, even by resorting to man-made laws under certain conditions. If all of that is impossible, then care must be taken to maintain communication with the daughter and raise her according to Islam, and to appease the mother to facilitate this. This matter requires wisdom and consultation with sensible people, and above all, seeking help from Allah and supplicating, and exerting effort to fulfill one's obligation.