Bad-tempered and sulky girl with American backing (6)
I believe the answer is:
sullen
'bad-tempered and sulky' is the definition.
(I've seen this before)
'girl with american backing' is the wordplay.
'girl' becomes 'nell' (girl's name).
'with' means one lot of letters go next to another.
'american' becomes 'US' (United States).
'backing' says the letters should be written backwards.
'nell'+'us'='nellus'
'nellus' back-to-front is 'SULLEN'.
(Other definitions for sullen that I've seen before include "Sulkily bad-tempered" , "Gloomy and bad-tempered" , "Resentful and unsociable in mood" , "Gloomy and petulant" , "Bad tempered, morose" .)