Core point rejected in half of Greece (7)
I believe the answer is:
essence
'core' is the definition.
(I know that core can be written as essence)
'point rejected in half of greece' is the wordplay.
'point' becomes 'ness' (I've seen this in other clues).
'rejected' shows that the letters should be reversed in order.
'in' is an insertion indicator.
'half of' means to take half the letters.
'ness' written backwards gives 'ssen'.
'greece' halved is 'ece'.
'ssen' placed inside 'ece' is 'ESSENCE'.
(Other definitions for essence that I've seen before include "Cologne, perhaps?" , "Flavour, spirit" , "Distinctive inner nature of person or thing" , "Intrinsic character" , "Intrinsic quality" .)