Claim to have worked around American show (7)
I believe the answer is:
musical
'show' is the definition.
(I know that show can be written as musical)
'claim to have worked around american' is the wordplay.
'to have worked' indicates an anagram (I've seen 'has worked' mean this (letters must be worked into new positions)).
'around' is an insertion indicator.
'american' becomes 'US' (United States).
'claim' is an anagram of 'mical'.
'mical' enclosing 'us' is 'MUSICAL'.
(Other definitions for musical that I've seen before include "Cats perhaps" , "Oklahoma, say" , "Anglo-Saxon tax for protection against invaders" , "Melodic" , "Drama with singing" .)