Not quite short of cheese (4)
I believe the answer is:
brie
'cheese' is the definition.
(I know that Brie is a type of cheese)
'not quite short' is the wordplay.
'not quite' means to remove the last letter (most but not all of the word).
'short' becomes 'brief' (similar in meaning).
'brief' with its final letter taken off is 'BRIE'.
'of' is the link.
(Other definitions for brie that I've seen before include "French cream cheese" , "Prov. squalid place" , "Soft, creamy French cheese" , "Variety of cheese" , "Round soft Fr. cheese" .)