Must a man from whom a married woman conceived—and who then married another man and bore him a child—acknowledge the child if it is his, and what are the implications of that?

Your friend must offer sincere repentance to Allah, cover himself, and cover the one with whom he committed the indecency. As for the child, he is not attributed to him, but rather to the woman's husband, because the Prophet, peace and blessings be upon him, said: "The child belongs to the bed, and for the fornicator is the stone."