What happens when a Ministorum Priest decides to go rogue and/or start to worship the Dark Gods? Extra heresy is what happens. Apostate Cardinals are members of the Ecclesiarchy who have since turned their back on the Emperor and the Imperial Creed. They are insane firebrands and rabble rousers, often turning the opinion and religious fealty of entire worlds with their anti-Imperial dogma and blasphemous sentiment. Apostate Cardinals are often accompanied by Pontifex Guards, grim adherents to their anti-Imperial ideology and fanatically loyal. They are considered as one of the worst and most dangerous enemies of the Ecclesiarchy as well as a colossal embarrassment as they can corrupt a shrine world from the inside out.
The most powerful of these was the Apostate Cardinal of Gathalamor during the Age of Apostasy, Bucharis, whose heresies reached such proportions they became known as the Plague of Unbelief. Yes, this was the fucker who invented this mindfuck of a disease. Not to be confused with the Curse of Unbelief, a potent arcane disease spread by servants of the Chaos God Nurgle in the 41st Millennium, better known as the Zombie Plague.
The role of the Apostate Cardinal is almost mirror that of their loyalist counterparts during times of war. Instead of using their rousing speech to purge the enemies of the Emperor, they instead inspire their fellow traitors with the litanies of hate and the destruction of the Emperor instead. If they fell to Chaos they would most likely be using Lorgar's Book of Chaos for both worship and daemon summoning.