Fragrance from pudding welcome after horrible starter (10)
I believe the answer is:
aftershave
'fragrance' is the definition.
(I've seen this before)
'pudding welcome after horrible starter' is the wordplay.
'pudding' becomes 'afters' (I've seen this before**).
'welcome' becomes 'ave' (I've seen this before).
'after' is a charade indicator (letters next to each other).
'starter' suggests taking the first letters.
The first letter of 'horrible' is 'h'.
'ave' after 'h' is 'have'.
'afters'+'have'='AFTERSHAVE'
'from' is the link.
(Other definitions for aftershave that I've seen before include "soothing application" , "Fragrance" , "alcohol to splash on, say" , "I'm slapped on the face" , "liquid splashed in bathroom" .)