Runs from 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 wordplay.
'british' becomes 'b' (abbreviation e.g. in 'BBC').
'city' becomes 'leeds' (Leeds is an example).
'from' acts as a link.
(Another definition for bleeds that I've seen is "Loses blood".)