Cousin must ignore American strike (4)
I believe the answer is:
coin
'strike' is the definition.
(both can mean to stamp metal to make coins)
'cousin must ignore american' is the wordplay.
'must ignore' suggests deleting specific letters.
'american' becomes 'us'.
'cousin' with 'us' taken out is 'COIN'.
(Other definitions for coin that I've seen before include "American quarter, for example" , "Invent (a new phrase)" , "Metal piece of money" , "Invent (a new word)" , "Think up" .)