Authority with no power to wield (5)

I believe the answer is:

exert

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'to wield' is the definition.
('exert' can be a synonym of 'wield')

I don't understand the remainder of the clue.

(Other definitions for exert that I've seen before include "Work very hard" , "Work strenuously" , "Bring into play, wield" , "Bring into use, apply" , "Apply (pressure)" .)

I've seen this clue in The Times.

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