All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad, sallallaahu 'alyhi wa sallam, is His slave and Messenger.
If by secret marriage you mean a man and a woman agreeing to get married without the consent of the woman’s guardian and the presence of witnesses, then this is Zinaa (fornication or adultery) and this is not permissible at all. However, if the marriage contract fulfilled the conditions and the pillars of a correct marriage, but it was not made public and it was kept secret from some people, then this marriage is valid and permissible according to the view of the majority of the scholars . For more benefit on the conditions of marriage, please refer to Fatwa 83629.
If the marriage which you mentioned was conducted in the manner that you mentioned, and the guardian was not present but the marriage registrar and the Shaykh who recited the Khutbah (sermon) could be considered as witnesses and they are eligible to be witnesses because of meeting the criteria of trustworthiness and the like, then such a marriage is valid according to the view of Imaam Abu Haneefah whereas it is considered as void according to the view of the majority of the scholars so it must be invalidated unless a judge who follows the Hanafi School of jurisprudence rules that it is valid or that the person who conducted the marriage follows the jurisprudence of Abu Haneefah . However, if there were no witnesses at the time of conducting the marriage contract or that the witnesses were dissolute people whose testimony is not valid, then such a marriage is void according to all four Schools of jurisprudence. In this case the sexual relationship based on such a marriage is nothing but Zinaa.
Therefore, both the young man and the girl are obliged to repent to Allaah for this matter and especially for the clear Zinaa that they had committed beforehand. Zinaa is a grave major sin and has its severe consequences in this world and in the Hereafter. Allaah Says (what means): {And do not approach unlawful sexual intercourse. Indeed, it is ever an immorality and is evil as a way.}[Quran 17:32]
Finally, if this young man wants to marry this girl, he may do so but he should first of all make sure that she is not pregnant according to the preponderant view of the scholars and then he may marry her according to the Sharee’ah with the consent of her guardian and the presence of two witnesses.
Allaah Knows best.