Capital consumption is breaking bailiff's heart (7)
I believe the answer is:
tbilisi
'capital' is the definition.
(capital of georgia)
'consumption is breaking bailiff's heart' is the wordplay.
'consumption' becomes 'tb' (consumption is a name for TB or tuberculosis).
'breaking' is an insertion indicator.
'heart' indicates the central letters.
The central letters of 'bailiff' are 'ili'.
'is' inserted inside 'ili' is 'ilisi'.
'tb'+'ilisi'='TBILISI'
(Other definitions for tbilisi that I've seen before include "Capital city of Georgia" , "Georgian place" , "Georgian city" , "Georgian capital" .)