Badly-behaved thug, a suspect in New York (7)
I believe the answer is:
naughty
'badly-behaved' is the definition.
(I've seen this before)
'thug a suspect in new york' is the wordplay.
'suspect' is an anagram indicator (a suspicious or unusual spelling).
'in' means one lot of letters goes inside another.
'new york' becomes 'ny'.
'thug'+'a'='thuga'
'thuga' is an anagram of 'aught'.
'aught' going into 'ny' is 'NAUGHTY'.
(Other definitions for naughty that I've seen before include "Mischievous - improper" , "Wicked" , "Badly-behaved" , "Behaving badly or risque" , "Badly behaved, mischievous" .)