What should a man do who committed Zina with a woman and fathered a child whom he then abandoned, then married her and fathered another child with her, in order to expiate for the sin of Zina and what he did to his first child?

The door of repentance is open to a servant as long as they have not reached the gurgling state [of death]. Repentance can transform bad deeds into good deeds, as Allah says: "Except for those who repent, believe and do righteous deeds. For them Allah will change their evil deeds into good ones." You must conceal your own sins and those of others, based on the hadith: "All of my Ummah will be forgiven except those who publicize their sins," and "Whoever conceals [the sin of] his Muslim brother in this world, Allah will conceal his [sins] on the Day of Resurrection." The mother must not sever ties with her son, but your paternity of him is not legitimate, and it is not permissible to adopt him. You could have used dissimulation (tawriyah) with your father to convince him that the story was false.