Is it permissible to marry and cohabit with a wife who browses pornographic websites, even after advising her about religion and prayer, and does the psychological distress resulting from this discovery affect the ruling on the marriage?

Watching pornographic films is forbidden (haram). If the woman is your wife, you must advise her by clarifying the ruling on what she watches. Encourage her to keep company with righteous women and attend gatherings of knowledge and remembrance, so that her faith may increase. Also, make supplication for her. It is advisable to complete the marriage with her as long as she is righteous and advises you to maintain the Fajr prayer. This might be a slip that can be remedied with advice and guidance.