Runs in British city (6)
I believe the answer is:
'runs' is the definition.
('bleed' can be a synonym of 'run')
'british city' is the wordplay.
'british' becomes 'b' (abbreviation e.g. in 'BBC').
'city' becomes 'leeds' (Leeds is an example).
'in' is the link.
(Another definition for bleeds that I've seen is "Loses blood".)