Do up short red coat by mistake (9)

I believe the answer is:

refurbish

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'do up' is the definition.
(thesaurus)

'short red coat by mistake' is the wordplay.
'short' means to remove the last letter.
'coat by' becomes 'fur' (fur is a kind of coat. I am not sure about the 'by' bit.**).
'mistake' becomes 'bish' (I've seen this in other clues).
'red' with its final letter taken away is 're'.
're'+'fur'+'bish'='REFURBISH'

Can you help me to learn more?

(Other definitions for refurbish that I've seen before include "Renovate, redecorate" , "Redecorate and brighten up" , "Clean up" , "Freshen up" , "Overhaul" .)

I've seen this clue in The Times.

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