Become old, caught in New York office (6)
I believe the answer is:
agency
'office' is the definition.
(synonyms)
'become old caught in new york' is the wordplay.
'become old' becomes 'age' (I've seen this in other clues).
'caught' becomes 'c' (cricket abbreviation).
'in' means one lot of letters goes inside another.
'new york' becomes 'ny'.
'c' inserted inside 'ny' is 'ncy'.
'age'+'ncy'='AGENCY'
(Other definitions for agency that I've seen before include "See 24 Across" , "Recruiters possibly" , "Government office" , "Bureau; instrumentality" , "Business that acts for other businesses" .)