In the UK, terrible puns are de rigeur, but the answer has to be a recognisable standalone word or phrase. So 'poisoned pen' could be an answer, and the clue could conceivably contain a pasta-based pun on that, but 'poisoned penne' could not.

In the US, at least (TIL there's a different style across the pond, so fun), a 'pun-based' clue like that would need a question mark at the end of the clue.

Additionally, I would say that that, because the answer isn't a coherent standalone word/phrase, that particular clue/answer combo would only be found in a puzzle as part of a hidden theme shared by a number of 'special clues' throughout the crossword, with some kind of rule uniting the answers (add 'ne' to the end of a common phrase, for example).

So the puns need to be in the clue, and not the answer?

Sort of. Cryptics, which are the international style, might have a clue like "insect in pants" -> "ant", which is pants without the first and last letters.

And more complicated variants, like anagrams, mashing words together, etc.

You could have a pun in the answer - although I think that's pretty rare - but the answer itself would still need to be a recognized word/phrase.

That's really interesting—I'm not much of a crosswords fan, but I had never even considered the idea that there are "dialects" in what crosswords clues & answers could be.