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This is a complex question, and I'll try to give an answer in two parts.

Unfortunately, this is a question that is asked a lot, but given that we don't know how long the relationship between Henry and Mary lasted, it is impossible to say. However, there are some clues that the relationship was common knowledge, enough that it damaged Henry's possibility of marrying Anne Boleyn later on. Henry had to apply for a dispensation to marry Anne, because he had been involved sexually with Mary. So that suggests not a hidden, perhaps one night stand as has been proposed, but something more public. Eric Ives, more of the foremost scholars on the Boleyn family, does not believe that either of Mary's children, Catherine, the elder, and Henry, were fathered by Henry, but based on timing, the favour shown to Mary's husband, and the favour shown to both children, I do not believe the possibility can be actively excluded, although I have strong doubts that Henry Carey, the younger, was the son of the king.

The most common answer to this question, 'no,' seems to rely on the fact that Henry did not acknowledge Mary's children as evidence they were not his. I don't agree with this assessment. Henry acknowledged Bessie Blount's son by him, and there were good reasons for doing so. Bessie was not married at the time of conception, and her affair with Henry was largely public.

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It was already largely known that the child was his. Further, as Henry Fitzroy got older, it became obvious that Henry would have no legitimate children with his wife Catherine. It was convenient to acknowledge the boy in case of no further sons appearing. None of the criteria that made acknowledging Henry Fitzroy desirable applied in Mary's case.

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Mary was already married, and had Henry acknowledged her children, they would have lost their status as the legitimate offspring of a powerful courtier and become bastards, being ineligible to inherit anything from their father's estates. Further, as they grew older, acknowledging them would have made it almost impossible to marry Anne, Mary's sister. The scandal would have been too great. One last point: if there were times when Mary was cohabiting with her husband AND conducting a sexual relationship with Henry, it is quite possible that none of the three involved knew whether the children were Henry's or not. One last point, although hardly decisive, is the favour shown to both children. Anne Boleyn took Henry Carey as her ward, ensuring him a prosperous future, and both Catherine and Henry Carey never fell out of royal favour. Catherine and Henry were both great favourites of Queen Elizabeth I, and Catherine's daughter Lettice resembled Elizabeth so deeply that she inadvertently caused offence.

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Sources: I refer to

The Life and Death of Anne Boleyn, E Ives - 2004, pp. 17

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TL;DR: It is impossible to say whether Catherine and Henry Carey were the children of Henry VIII; there is conflicting evidence, but the possibility cannot be excluded based on Henry's lack of acknowledgment of both children