An obsession about New York, no matter what (8)
I believe the answer is:
anything
'no matter what' is the definition.
(both can mean any item)
'an obsession about new york' is the wordplay.
'an' becomes 'a'.
'obsession' becomes 'thing' (I've seen this before).
'about' is a charade indicator (letters next to each other).
'new york' becomes 'ny'.
'thing' after 'ny' is 'nything'.
'a'+'nything'='ANYTHING'
(Other definitions for anything that I've seen before include "Musical by 14" , "Aught" , "Unspecified object" , "An unspecified item" , "- - Goes (mus.)" .)