What may need tightening in the City possibly or leafy suburbia? (11,4)
I believe the answer is:
stockbroker belt
'leafy suburbia?' is the definition.
(term for the suburbs of a city)
I can't explain the rest of the clue.
(Other definitions for stockbroker belt that I've seen before include "wealthy area" , "part of Surrey?" , "Affluent residential area" , "Wealthy neighbourhood" , "Prosperous area" .)