Cut fabric that’s fine grade (8)

I believe the answer is:

lacerate

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'cut' is the definition.
(I've seen this before)

'fabric that's fine grade' is the wordplay.
'fabric' becomes 'lace' (lace is a kind of fabric).
'that's' says to put letters next to each other.
'fine grade' becomes 'rate' (synonyms. I am not sure about the 'fine' bit.).
'lace'+'rate'='LACERATE'

Can you help me to learn more?

(Other definitions for lacerate that I've seen before include "Tear, rip (flesh or skin)" , "Cut, leaving jagged edges" , "Deeply cut (flesh, skin)" , "Cut or tear irregularly" , "Clear tea (anag.)" .)

I've seen this clue in The Sunday Times.

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