Determination to improve sodden eggs (10)
I believe the answer is:
doggedness
'determination' is the definition.
(I know that doggedness is a type of determination)
'improve sodden eggs' is the wordplay.
'improve' indicates anagramming the letters.
'sodden'+'eggs'='soddeneggs'
'soddeneggs' is an anagram of 'DOGGEDNESS'.
'to' is the link.
(Other definitions for doggedness that I've seen before include "Tenacity" , "Determination" , "Persistence" .)