American soldier going after my dog (5)
I believe the answer is:
corgi
'dog' is the definition.
(corgi is a kind of dog)
'american soldier going after my' is the wordplay.
'american soldier' becomes 'gi'.
'going after' means one lot of letters go next to another.
'my' becomes 'cor' (I've seen this in other clues).
'gi' after 'cor' is 'CORGI'.
(Other definitions for corgi that I've seen before include "One of the Queen's dogs?" , "man's best friend" , "Welsh breed of short-legged dog" , "Royal pet" , "The Queen's favourite?" .)