What is the expiation for engaging in foreplay with a girl that led to her claiming pregnancy, and what is the ruling on the child if his paternity is not established, and what are the implications for the husband if the child’s paternity is established and he desires a divorce?

You must repent to Allah for what happened with the woman, by abandoning the sin, regretting it, and resolving not to return to it, while also maintaining concealment.

Your marriage to the woman before she gives birth, if she is pregnant, is invalid according to some scholars and valid according to others.

As for the lineage of the pregnancy: it is not attributed to you if the birth occurs less than six months after your marriage. It is attributed to you if the birth occurs more than six months after your marriage, and you cannot deny its lineage except through li'an (oath of condemnation). DNA analysis is not relied upon for denying lineage.

If the child's lineage is disproven from you, he has no right over you. However, if his lineage is not disproven from you, then he has the rights of a child over his father, such as financial support and upbringing.