Runs in British city (6)
I believe the answer is:
Here is my best explanation:
'runs' is the definition.
('bleed' can be a synonym of 'run')
'british city' is the subsidiary indication.
'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".)