Time to get caught in city office (6)

I believe the answer is:

agency

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'city office' is the definition.
'agency' can be an answer for 'office' (synonyms). I'm not sure about the 'city' bit.

'time to get caught' is the wordplay.
'time' becomes 'age' (an age is a period of time).
'to' becomes 'ny' (I can't explain this - if you can you should believe this answer much more).
'get' is an insertion indicator.
'caught' becomes 'c' (cricket abbreviation).
'ny' going around 'c' is 'ncy'.
'age'+'ncy'='AGENCY'

'in' is the link.

Can you help me to learn more?

(Other definitions for agency that I've seen before include "Recruiters possibly" , "Bureau finding jobs etc" , "Means - business" , "The A in CIA" , "Business that works on behalf of others" .)

I've seen this clue in The Mirror.

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