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.
'twelve' becomes 'dozen' (synonyms).
'dozen' with its final letter removed is 'DOZE'.
'it's' acts as a link.
(Other definitions for doze that I've seen before include "Sleep lightly" , "A short light sleep" , "Sleep fitfully" , "little sleep" , "Take forty winks" .)