How can a man atone for his sin after having mentioned the name of an innocent girl to justify his calls with other women, which led to the destruction of her reputation and caused the death of his son and the illness of his wife and daughter as a result of the girl's supplication against him?

The man must repent sincerely to Allah Almighty for slandering the innocent woman. He must seek forgiveness for her and obtain her absolution for the harm he inflicted upon her, if possible. He should also clarify her innocence and admit that he lied about her to those whose perception of her has been tainted.