A street in New York that's unpleasant (5)
I believe the answer is:
nasty
'unpleasant' is the definition.
(nasty can mean unpleasant or horrible)
'a street in new york' is the wordplay.
'street' becomes 'st' (abbreviation).
'in' is an insertion indicator.
'new york' becomes 'ny'.
'a'+'st'='ast'
'ast' inserted into 'ny' is 'NASTY'.
'that's' acts as a link.
(Other definitions for nasty that I've seen before include "Highly repugnant" , "Foul" , "rank" , "Evil" , "Offensive or malicious" .)