Begrudge living expense around south-east (6)
I believe the answer is:
resent
'begrudge' is the definition.
(synonyms)
'living expense around south-east' is the wordplay.
'living expense' becomes 'rent' (I've seen this in another clue).
'around' is an insertion indicator.
'south east' becomes 'SE' (abbreviation).
'rent' placed around 'se' is 'RESENT'.
(Other definitions for resent that I've seen before include "Dispatched again - take amiss" , "Object to" , "Dislike" , "Nurse bitter or angry feeling" , "Feel aggrieved at" .)