How can one deal with a husband who is negligent in performing Fajr prayer and oversleeps through some prayers due to lethargy, and then performs them after waking up without showing distress over missing their prescribed times, despite attempts to persuade him that were met with rejection, taking into account the deterioration of the marital relationship and the wife's consideration of divorce in order to ensure the religious upbringing of their son?

We advise you to continue counseling your husband and reminding him of the immense importance of prayer and the necessity of performing it on time, using a gentle approach, and not to despair of his improvement. It is best for a woman to remain in her home and not go out unless there is a need. You are permitted to seek divorce (Talaq) or annulment (Khul') if he persists in his negligence of prayer and life remains strained, after performing Istikhara (prayer for guidance), seeking counsel, and making supplication.