Is it permissible for a mother to engage in an illicit relationship with a man who is a stranger to her after her separation from her husband, receiving love letters and pictures from him?

We advise you to strive in counseling your mother gently, reminding her of Allah, and frightening her with the consequences of disobeying Him. Also, supplicate for her frequently. It would be commendable if you sought to reconcile her with her husband. If that does not help and she remains in her state, then inform someone who can prevent her from this evil, and you must honor her and treat her kindly, no matter her condition.