A broom was a type of cleaning implement, with a brush and a handle. (TV: Deep Breath) It was also associated with magic and witchcraft. (WC: New Adventures Await!, PROSE: Hervoken Halloween [+]James Hornby, Horrors of Arcbeatle (SIGNET, Arcbeatle Press, 2023)., etc.)
Harry Potter once flew through New York City on a broom. (WC: New Adventures Await!)
For Halloween, SIGNET member Aoife Fitzgerald's young daughter Norah once dressed up as a witch, complete with a plastic broomstick and a toy wand. The broomstick was later genuinely, if briefly, levitated by Hervy the Hervoken, who used it to bludgeon a knife out of an attacking Aoife's hand. (PROSE: Hervoken Halloween [+]James Hornby, Horrors of Arcbeatle (SIGNET, Arcbeatle Press, 2023).)
Tosh once stole some corn, a broom and a few other things from the Third Doctor's greenhouse. (PROSE: Morphology)
One time when the Tenth Doctor visited Earth he found it was missing. There was only a white void with a small, old house being defended by Bob Manning with a broom from dragon-like creatures. (COMIC: The House at the End of the World)
In Venice in 1580, Rory Williams tried to fight Francesco with a broom. (TV: The Vampires of Venice)
The Twelfth Doctor used the idea of the parts of a broom being constantly replaced to explain to the Half-Face Man how it had rebuilt itself so many times it was no longer its original self. (TV: Deep Breath)
During the Twelfth Doctor's imprisonment in the Prison, Clara Oswald borrowed a broom from Danny Pink so that, with the help of Classes 2A and 2B, she could make a placard with the message "FREE THE DOCTOR". (PROSE: The Blood Cell)