Run-down saloon perhaps causing bishop great displeasure (6)
I believe the answer is:
banger
'run-down saloon perhaps' is the definition.
(banger can mean an old car)
'bishop great displeasure' is the wordplay.
'bishop' becomes 'B' (chess abbreviation).
'great displeasure' becomes 'anger' (anger is displeasure or annoyance).
'b'+'anger'='BANGER'
'causing' acts as a link.
(Other definitions for banger that I've seen before include "Noisy old car or slangy sausage" , "Slangily, a noisy old car or a sausage" , "Old car (colloq.)" , "Sausage (informal)" , "Noisy old car or slang sausage" .)