Poem found after graduate goes back home (5)
I believe the answer is:
abode
'home' is the definition.
(synonyms)
'poem found after graduate goes back' is the wordplay.
'poem' becomes 'ode' (ode is a type of poem).
'found after' says to put letters next to each other.
'graduate' becomes 'BA' (someone with a Bachelor of Arts degree).
'goes back' shows that the letters should be reversed in order.
'ba' reversed gives 'ab'.
'ode' after 'ab' is 'ABODE'.
(Other definitions for abode that I've seen before include "House made of adobe" , "stay" , "Archaic dwelling place" , "Place of residence" , "Dwelling, residence" .)