Cutting a new sort of woollen cloth (6)

I believe the answer is:

tartan

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'sort of woollen cloth' is the definition.
'tartan' can be an answer for 'cloth' (tartan is a kind of cloth). I'm not sure about the 'sort of woollen' bit.

'cutting a new' is the wordplay.
'cutting' becomes 'tart' (both can mean sharp or sarcastic).
'new' becomes 'n' (common abbreviation eg NT for New Testament).
'tart'+'a'+'n'='TARTAN'

Can you help me to learn more?

(Other definitions for tartan that I've seen before include "Criss-cross pattern" , "Cloth with a plaid design" , "Distinctive cloth pattern of a Highlands clan" , "Woollen checked material" , "Scottish plaid design" .)

I've seen this clue in The Times.

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