District of capital, heart of Baghdad (6)
I believe the answer is:
parish
'district' is the definition.
(parish is a kind of district)
'capital heart of baghdad' is the wordplay.
'capital' becomes 'paris' (Paris is an example).
'heart of' means to look at the middle letters.
The middle of 'baghdad' is 'h'.
'paris'+'h'='PARISH'
'of' acts as a link.
(Other definitions for parish that I've seen before include "Local church community" , "District with own church and clergyman" , "quarter" , "Rector's responsibility" , "In Louisiana a county" .)