Why did Henry VIII come into conflict with the papacy in the early 1530s?

This is a good question, and one that has multiple overlapping causes. To begin with, Henry's wishes clashed with the papacy's as he wanted to divorce Catherine of Aragon, in order to produce a legitimate male heir. This wish to divorce Catherine contravened traditional papal ideas of the sanctity of marriage, and so required special permission from the pope. However, this permission was unlikely to be granted, as Pope Clement VII was under the control of the Holy Roman Emperor Charles V after Charles' conquest of northern Italy in 1527, and Charles was unwilling to see his relative, Catherine, humiliated by divorce.More than this, Henry's personality clashed with papal authority, as he increasingly saw himself as deserving of more power over religious matters. He was encouraged in this by his Protestant-inspired chief minister, Thomas Cromwell, as well as his own egotism and desire for power. Henry pursued his own theological justifications for the divorce, citing Catherine's previous marriage to his brother as rendering the marriage null and void due to a passage from Leviticus, and became increasingly convinced by his own reasoning. A final straw in this conflict was the emergence of Anne Boleyn as Henry's mistress and then his wife in 1532, which led Henry to become more desperate in his efforts to obtain a divorce, paving the way for the split from Rome.