Determined to cook a bad egg, note (6)
I believe the answer is:
dogged
'determined' is the definition.
(I've seen this before)
'cook a bad egg note' is the wordplay.
'cook' becomes 'do' (do can mean to cook).
'a bad' indicates anagramming the letters.
'note' becomes 'd' (musical note).
'egg' with letters rearranged gives 'gge'.
'do'+'gge'+'d'='DOGGED'
'to' is the link.
(Other definitions for dogged that I've seen before include "Resolute; trailed" , "Tenacious, persistent" , "Stubborn or determined in canine manner" , "Persistent in a canine sort of way" , "Stubbornly persisting" .)