Unsteady Byron consuming fine port in part of New York City (8)
I believe the answer is:
brooklyn
'part of new york city' is the definition.
('Brooklyn' is part of 'new york city')
'unsteady byron consuming fine port' is the wordplay.
'unsteady' is an anagram indicator.
'consuming' means one lot of letters goes inside another.
'fine' becomes 'ok' ('OK' can be a synonym of 'fine').
'port' becomes 'l'.
'byron' anagrammed gives 'broyn'.
'ok'+'l'='okl'
'broyn' going around 'okl' is 'BROOKLYN'.
'in' is the link.
(Other definitions for brooklyn that I've seen before include "New York area" , "part of Long Island?" , "District and bridge in New York" , "Big Apple borough" , "NYC borough, book and film" .)