Food shop fine? Not half! (4)
I believe the answer is:
deli
'food shop' is the definition.
(short for delicatessen)
'fine? not half' is the wordplay.
'fine?' becomes 'delicate' ('delicate' can be a synonym of 'fine').
'not half' indicates taking half.
'delicate' with half the letters taken is 'DELI'.
(Other definitions for deli that I've seen before include "Shop selling foreign foods" , "Specialist food shop" , "shop briefly" , "Food store" , "Food shop or counter" .)