1) It is quite possible that his mother is married. However, Rashi (DH v'Eino) writes that if the child calls a man "Aba" the mother silences him. It could be that it is the mother's husband that the child tries to call his father.

2) However, the more usual scenario would likely be that it is well-known that the mother is unmarried. This would be similar to the Mishnah in Kesuvos 13a, which discusses a single woman who is pregnant. They asked her who is the father of the baby and she told them his name and claimed that he is a Kohen (this does not literally mean that he is a Kohen, but rather that he is of good pedigree and is not a Mamzer). Raban Gamliel and Rebbi Eliezer say that we believe her, and the Halachah follows this opinion.

3) We learn from here that, generally speaking, it is more important what the mother knows than what the child himself knows. In fact, it would often be difficult to accept the testimony of the child. If he is a minor, he is not a valid witness, and even if he is an adult the onus of proving who the father is would fall upon him.

4) The Rambam (Hilchos Isurei Bi'ah 15:12) writes that if we were unable to receive the testimony of the mother, or if she admitted that she had relations with a Mamzer, the child would be a Safek Mamzer, because we would say that just as she admitted having relations with this man, she might possibly have had relations with another. The Rambam concludes that this child is termed a "Shetuki" who knows who is mother is but does not know for sure who his father is.

5) We learn from the Rambam that the chief problem with the Shetuki is that the child cannot know for sure who his father is. If he would be able to prove that his father is Kosher, then he could marry any Jewish girl, but the Shetuki is a Safek Mamzer

Kol Tuv,

Dovid Bloom