Why plan for Mary to give the King a son?

Henry bestowed importance and wealth upon his illegitimate sons; it would have kept them in his favor forever. He only sent the child away because Anne promised him legitimate sons.

I just don't understand the point of this part of the film.

Henry was desperate enough for a son to actively consider making his eldest bastard Henry Fitzroy his heir. It never happened, as the 3rd wife gave him a legitimate son, but it was under consideration if there were no male heirs*, as a sort if emergency backup desperate measure to avoid chaos.

Edward Henry's son died very young, of course, and his heirs were his sisters Mary and Elizabeth. Both of whom had been declared bastards when they were younger, so yes, in a way it was possible for a bastard of Henry's to rule.

* Of course Henry had a legitimate daughter.