Depressing experience after star offers drink (9)
I believe the answer is:
sundowner
'drink' is the definition.
(I've seen this before)
'depressing experience after star' is the wordplay.
'depressing experience' becomes 'downer' (I've seen this before).
'after' means one lot of letters go next to another.
'star' becomes 'sun' (Sun is an example).
'downer' put after 'sun' is 'SUNDOWNER'.
'offers' is the link.
(Other definitions for sundowner that I've seen before include "drink after work?" , "cocktail?" , "Evening draught" , "Evening drink" , "Twilight drink" .)