The influence, at some stage, of a fancy piece? (6)

I believe the answer is:

agency

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'stage of a fancy piece?' is the definition.
The definition and answer can be both states of affairs as well as being singular nouns.
Maybe they are linked in a way I don't understand?

I don't understand how the remainder of the clue works.

Can you help me to learn more?

(Other definitions for agency that I've seen before include "Administrative unit" , "Travel --; dating --" , "Type of business" , "Business that conducts business on behalf of another" , "Business that acts for other businesses" .)

I've seen this clue in the Evening Standard.

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