All perfect praise be to Allah, The Lord of the Worlds. I testify that there is none worthy of worship except Allah, and that Muhammad is His slave and Messenger.
This man has Islamically no relation with that child and there is no blood relationship between them. His passion and love towards that child is a sign that he is not sincere in his repentance and that he did not sincerely regret committing Zina (fornication or adultery) with the mother of that child. If he had really regretted what he did, he would have felt sorrowful whenever he sees that child out of fear that Allah would punish him for that sin.
Therefore, this man should sincerely repent to Allah and seek forgiveness and perform good deeds as much as possible. Also, he should stay far away from the mother of that child as the devil might push him to commit the same sin again with her. Besides, he has no right to ask after that child or to ask that he stays with him because he has Islamically no relationship with him as we previously mentioned.
Allah knows best.