I've only heard of pre-Hanover Kings having illegitimate children, did any modern monarch act similarly, or are they just doing a better job of covering up?
The last monarch known for sure to have illegitimate children was William IV (r. 1830-1837), who had 10 children with his mistress Dorothea Jordan before marrying his wife and eventual queen Adelaide of Saxe-Meiningen. These were given the surname FitzClarence (William was Duke of Clarence at the time). There was also at least one son conceived before he met Jordan whose mother is unknown.
I say 'last monarch known for sure' because Edward VII (r. 1901-1910) was a notorious playboy with various mistresses, many of whom married and bore children of their own; several of these have been speculated to be Edward's children, but beyond inferences such as 'she had the child around the same time she was seeing the Prince' there's no solid proof.
For more detail on William and his relationships consider William IV by Philip Ziegler; a good general reference book on royal family trees is Britain's Royal Families: The Complete Geneology by Allison Weir.
Semi-related: how well recorded are cases of monarchs having illegitimate children? I mean do we know about most of them, or were there probably a lot more than we know about?