Is it permissible to continue a relationship with a young man who drinks alcohol and has sexual relations, with the intention of guiding him, and are there specific supplications for that?

It is not permissible, شرعًا (religiously), to establish a relationship between a man and a woman outside the framework of marriage. Allah has forbidden handshakes between the sexes and خلوة (private seclusion), and has forbidden looking [at what is unlawful], enjoining the lowering of the gaze, as evidenced by the Hadith: "For one of you to be stabbed in the head with an iron needle is better for him than to touch a woman who is not lawful for him," and the Hadith: "No one of you should seclude himself with a woman, for indeed Satan will be the third party between them," and the noble verse: "قُلْ لِلْمُؤْمِنِينَ يَغُضُّوا مِنْ أَبْصَارِهِمْ وَيَحْفَظُوا فُرُوجَهُمْ ذَلِكَ أَزْكَى لَهُمْ إِنَّ اللَّهَ خَبِيرٌ بِمَا يَصْنَعُونَوَقُلْ لِلْمُؤْمِنَاتِ يَغْضُضْنَ مِنْ أَبْصَارِهِنَّ وَيَحْفَظْنَ فُرُوجَهُنَّ وَلا يُبْدِينَ زِينَتَهُنَّ إِلَّا مَا ظَهَرَ مِنْهَا" (Tell the believing men to reduce [some] of their gaze and guard their private parts. That is purer for them. Indeed, Allah is Acquainted with what they do. And tell the believing women to reduce [some] of their gaze and guard their private parts and not expose their adornment except that which [necessarily] appears thereof).