The Greek bread is mine, thanks (5)
I believe the answer is:
pitta
'the greek bread' is the definition.
(I've seen this before)
'mine thanks' is the wordplay.
'mine' becomes 'pit' (down the pit).
'thanks' becomes 'ta' (UK informal version of 'thank you').
'pit'+'ta'='PITTA'
'is' is the link.
(Other definitions for pitta that I've seen before include "Flat bread that's split to take a filling" , "Flat sort of bread" , "Type of flat bread that's split to hold a filling" , "Flat hollow unleavened bread" , "(E.g. Greek) bread" .)