After study put up companion in private Italian course (7)
I believe the answer is:
gnocchi
'italian course' is the definition.
(Italian dish)
'after study put up companion in private' is the wordplay.
'after study' becomes 'con' (conning is a kind of studying. I am not sure about the 'after' bit.).
'put up' shows that the letters should be reversed in order.
'companion' becomes 'ch' (Companion of Honour).
'in' is an insertion indicator.
'private' becomes 'gi' (GI can informally mean a US soldier).
'con' in reverse letter order is 'noc'.
'noc'+'ch'='nocch'
'nocch' going inside 'gi' is 'GNOCCHI'.
(Other definitions for gnocchi that I've seen before include "Pasta or potato dumplings" , "Italian dish" , "simple meal?" , "Small Italian dumplings similar to pasta" , "Dumpling dish" .)