Cut cost - on a shoestring (8)
I believe the answer is:
lacerate
'cut' is the definition.
(thesaurus)
'cost on a shoestring' is the wordplay.
'cost' becomes 'rate' (rate is a kind of cost).
'on' is a charade indicator (letters next to each other).
'a shoestring' becomes 'lace' (I've seen this before).
'rate' put after 'lace' is 'LACERATE'.
(Other definitions for lacerate that I've seen before include "Injure be tearing, wound roughly" , "Clear tea (anag.)" , "Harrow" , "Tear, rip (flesh or skin)" , "Tear, wound, hurt" .)