One giving the chimney a brush up? (5)

I believe the answer is:

sweep

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'brush up?' is the definition.
'sweep' can be an answer for 'brush' (synonyms). I am not sure about the 'up?' bit.

I can't explain the remainder of the clue.

Can you help me to learn more?

(Other definitions for sweep that I've seen before include "clear" , "Brush clean" , "if clean, could result in a whitewash" , "Long oar" , "Clean with broom" .)

I've seen this clue in the Evening Standard.

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