Fine Greek character offers cheese (4)
I believe the answer is:
feta
'cheese' is the definition.
(type of cheese)
'fine greek character' is the wordplay.
'fine' becomes 'f' (abbreviation in pencil classifications).
'greek character' becomes 'eta' (letter from the Greek alphabet).
'f'+'eta'='FETA'
'offers' acts as a link.
(Other definitions for feta that I've seen before include "Greek cheese from goat or sheep milk" , "Low-fat cheese" , "food there" , "A ewe's, goat's milk cheese" , "White Greek cheese" .)