Part of London that's not quite so warm (4)
I believe the answer is:
soho
'part of london' is the definition.
(district of Westminster)
'not quite so warm' is the wordplay.
'not quite' means to remove the last letter (most but not all of the word).
'warm' becomes 'hot' (associated in meaning).
'so'+'hot'='sohot'
'sohot' with its last letter taken off is 'SOHO'.
'that's' is the link.
(Other definitions for soho that I've seen before include "Central London night life district" , "Theatre and night-life district of London" , "Hunter's announcement of sighting a hare" , "fashionable neighbourhood (or neighborhood)" , "Once notorious part of central London" .)