After period of time, client initially entered New York office (6)
I believe the answer is:
agency
'office' is the definition.
('agency' can be a synonym of 'office')
'after period of time client initially entered new york' is the wordplay.
'after' means one lot of letters go next to another.
'period of time' becomes 'age' (age is a kind of period of time**).
'initially' says to take the initial letters.
'entered' means one lot of letters goes inside another.
'new york' becomes 'ny'.
The initial letter of 'client' is 'c'.
'c' put into 'ny' is 'ncy'.
'age' put next to 'ncy' is 'AGENCY'.
(Other definitions for agency that I've seen before include "Government unit - business providing services" , "Travel --; dating --" , "Business that works on behalf of others" , "Recruitment bureau" , "Office or business of one who does business for others" .)