Have a nap? It's not quite twelve! (4)
I believe the answer is:
doze
'have a nap?' is the definition.
(I've seen this before)
'not quite twelve' is the wordplay.
'not quite' means to remove the last letter (most but not all of the word).
'twelve' becomes 'dozen' (synonyms).
'dozen' with its final letter taken away is 'DOZE'.
'it's' is the link.
(Other definitions for doze that I've seen before include "Sleep lightly" , "A short light sleep" , "Sleep fitfully" , "little sleep" , "Take forty winks" .)