Man wishing each woolly could be cleaner (7,7)

I believe the answer is:

washing machine

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'be cleaner' is the definition.
The answer and definition can be both man-made objects as well as being singular nouns.
Maybe they are linked in a way I don't understand?

'man wishing each woolly' is the wordplay.
'man wishing each' can be anagrammed to 'WASHING MACHINE'.
Though, I am not sure how this is indicated.

'could' is the link.

Can you help me to learn more?

I've seen this clue in The Telegraph.

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