Bad painting of a bud, possibly (4)
I believe the answer is:
daub
'bad painting' is the definition.
(term for an unprofessional painting)
'a bud possibly' is the wordplay.
'possibly' indicates an anagram.
'a'+'bud'='abud'
'abud' with letters rearranged gives 'DAUB'.
'of' is the link.
(Other definitions for daub that I've seen before include "Plaster; paint crudely" , "Do graffiti" , "coat casually" , "Smear carelessly" , "Amateurish painting" .)