Pet left annoying American (7)
I believe the answer is:
doggone
'annoying american' is the definition.
(US word similar to 'damned' used when annoyed)
'pet left' is the wordplay.
'pet' becomes 'dog'.
'left' becomes 'gone' (I've seen this before).
'dog'+'gone'='DOGGONE'
(Other definitions for doggone that I've seen before include "Infernal" , "expression used by irritated person" .)