Is a husband's customary marriage to a divorcée, without her family's knowledge, Islamically valid, and what is the expiation for adultery in this case?

The husband committed a grave sin: adultery. His marriage contract with that woman is invalid for several reasons: he married her while she was in her 'iddah (waiting period), there was no guardian (wali), and there was a deficiency in witnesses, as the husband himself cannot be a witness. Furthermore, there is a scholarly dispute regarding the validity of the adulterer's marriage before repentance, and concerning the perpetual prohibition of marrying a woman during her 'iddah. Therefore, he must abstain from that woman and sever his relationship with her. If he has a child with her, the lineage of that child is attributed to him due to his belief in the validity of the marriage. He must engage in sincere repentance and seek Allah's concealment.