Songbird Greek character buried in mine (5)
I believe the answer is:
pipit
'songbird' is the definition.
(I've seen this before)
'greek character buried in mine' is the wordplay.
'greek character' becomes 'pi' (letter from the Greek alphabet).
'buried in' is an insertion indicator.
'mine' becomes 'pit' (down the pit).
'pi' going into 'pit' is 'PIPIT'.
(Other definitions for pipit that I've seen before include "Titlark" , "Small bird of rock or meadow" , "Songbird of open country" , "Songbird with streaky brown plumage" , "Lark-like songbird" .)