Cheese crisp? Not quite (4)
I believe the answer is:
'cheese' is the definition.
(I've seen this before)
'crisp? not quite' is the wordplay.
'crisp?' becomes 'brief' ('brief' can be a synonym of 'crisp').
'not quite' means to remove the last letter.
'brief' with its final letter removed is 'BRIE'.
(Other definitions for brie that I've seen before include "Soft and creamy mild white cheese", "Soft, creamy French cheese", "French 8 across", "cheesy stuff", "Prov. squalid place".)