Exhaust broken by stick (4,4)

I believe the answer is:

wear down

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'exhaust broken by stick' is the definition.
'wear down' can be an answer for 'exhaust' (thesaurus). I am unsure of the 'broken by stick' bit.

'exhaust broken by' is the wordplay.
'exhaust' becomes 'wear' (I've seen this before).
'broken' becomes 'down' (both can mean not working).
'by' means one lot of letters go next to another.
'wear'+'down'='WEAR DOWN'

Can you help me to learn more?

(Another definition for wear down that I've seen is " Gradually erode".)

I've seen this clue in The Guardian.

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