Can a woman prove marriage through common knowledge, knowing that the man acknowledged paternity of her children but denied marriage to her? And is his acknowledgment of the children and her claim of marriage sufficient to prove the marriage?

Testimony to marriage by widespread repute is valid according to most scholars, and marital status is established by acknowledging a child of the woman. If the matter is brought before the Sharia (definition) court, it will rule on it based on the evidence presented.