Bury's season failing to start (5)
I believe the answer is:
inter
'bury's' is the definition.
('inter' can be a synonym of 'bury')
'season failing to start' is the wordplay.
'season' becomes 'winter' (winter is a kind of season).
'failing to start' means to remove the first letter.
'winter' with its initial letter taken off is 'INTER'.
(Other definitions for inter that I've seen before include "promoting cessation" , "Set deep" , "Milan FC" , "Put under the earth" , "Nitre (anag.)" .)