The phrase "a pun based on" is correct and usable in written English.
You can use it when explaining the origin or nature of a pun, indicating that it derives from a specific word or phrase.
Example: "The joke was a pun based on the word 'bark,' referring both to tree bark and the sound a dog makes."
Alternatives: "a play on words involving" or "a wordplay centered around".
It's a pun based on the MPs' house expenses scandal.
"Qui Trop Embrasse," whose title literally means "Who kisses too much," is a pun based on a French proverb that cautions against overreaching, and in both senses it's apt for the movie's tone of breathlessly intimate striving.
Even if there's gunfire everywhere, stop everything you're doing after a notable kill to make a pun based on their death.
The episode title, which is often wrongfully referred to as "The Latest Gun In the West", is a pun based on the term "the fastest gun in the west".
Ideally it is a pun based on a fandom you enjoy.
Never make a pun based on any aspect of her personality/appearance, at least until you get to know her better.
But Eris has a moon, Dysnomia, meaning lawlessness, a subtle pun based on the old nickname Xena, the "Warrior Princess" character played by Lucy Lawless on TV.
I've reflected on how sometimes the funniest people write the deepest things; we shouldn't write anyone off (excuse the pun), based on personality.
As one interpreter has noted in an academic article, "Puns based on a single word with multiple meanings in the source language should generally not be attempted by interpreters, as the result will probably not be funny".
The titles of the first five episodes, "Captain Cook", "Corporal Punishment", "Major Star", "Private Plane" and "General Hospital" are puns based on the pairing of a military rank and another word related to the episode's content.
Wow your family and friends with some pre-cooked headline puns based on the vice-presidential candidates' names.
