Forty winks? Just under twelve (4)
I believe the answer is:
doze
'forty winks?' is the definition.
(I've seen this before)
'just under twelve' is the wordplay.
'just under' means to remove the last letter (the majority but not all of the word is written).
'twelve' becomes 'dozen' (synonyms).
'dozen' with its final letter removed is 'DOZE'.
(Other definitions for doze that I've seen before include "Go to sleep" , "Snooze" , "A short light sleep" , "Sleep fitfully" , "Take nap" .)