Girl appearing in An American Dream (6)
I believe the answer is:
andrea
'girl' is the definition.
'in an american dream' is the wordplay.
'in' indicates the answer is hidden within the clue.
'ANDREA' can be found hidden inside 'an american dream'.
'appearing' is the link.
(Other definitions for andrea that I've seen before include "Girl's (in Italy, boy's) name" , "Name of English girl or Italian boy" , "Name for British girl or Italian boy" .)