Factual error: In 1638, a mathematician calculates that the asteroid will return in 350 years time, on the 23rd of November 1988. But in 1752, Britain's calendar was moved forward 11 days so that it matched with that of the Vatican. This means the asteroid would return on 4th December, so everything that is supposed to happen in this story after 1752 is 11 days out.
Plot hole: In episode 1, on both occasions that Silas P presents his card (first to Daphne S, then later to the Doctor), he gives it to his victims the wrong way around. They have to surreptitiously turn it over so that the side with his name on it is facing them, and then act surprised when he tells them to flip the card over to reveal the words "Happiness Patrol Undercover", as if they hadn't seen it before.