Grow old and cold in New York office (6)
I believe the answer is:
agency
'office' is the definition.
('agency' can be a synonym of 'office')
'grow old and cold in new york' is the wordplay.
'grow old' becomes 'age' ('age' can be a synonym of 'grow old').
'and' says to put letters next to each other.
'cold' becomes 'c' (eg on taps).
'in' is an insertion indicator.
'new york' becomes 'ny'.
'c' put inside 'ny' is 'ncy'.
'age'+'ncy'='AGENCY'
(Other definitions for agency that I've seen before include "Function of one who acts for another" , "Business that acts for another business" , "Business that conducts business on behalf of another" , "Business that acts for other businesses" , "Business that represents actors etc." .)