Your teacher is probably referring to the Duke of Richmond, Henry FitzRoy, who was essentially the only one of Henry VIII's reputed illegitimate sons to be acknowledged by the king. In his youth the young FitzRoy was made a Duke and, throughout his life was bestowed several higher honours, at one point even being considered for the Crown of the Ireland though Henry's Chamberlain advised against the idea of creating a separate kingdom in an already restless area and it was stricken down.

As far as considering FitzRoy as his heir, we can only speculate as to whether or not Henry VIII entertained this thought seriously though we do know that the Second Succession Act (which was passed) would have allowed the King to select his own successor, leaving the logical choice for that selection to be FitzRoy unless the King had begotten a legitimate son.

However, Henry FitzRoy died under somewhat vague conditions in the very year that Act was passed.

I'll leave someone else to comment with more evidence as to whether the court or country would have accepted FitzRoy's succession, though with the heavy turmoil and political intrigue that surrounded Henry VIII and his successors, there's probably no clear answer.

Source: FitzRoy's biographer, Beverly Murphy's The bastard prince: Henry VIII’s lost son

There's quite a bit of information about this in Alison Weir's biography of Henry. For example, she states, “It was not unusual for princes to advance their bastards, but there was little doubt that this boy was being groomed for kingship. He was “well brought up like a prince’s child” and kept “the state of a great prince.” He held court from a rich chair of cloth of gold set under a canopy of estate, and was addressed as royalty.”

Excerpt From: Weir, Alison. “Henry VIII.”