Crone a non-starter in desire (4)
I believe the answer is:
itch
'desire' is the definition.
(I know that itch is a more specific form of the action desire)
'crone a non-starter' is the wordplay.
'crone' becomes 'witch' (synonyms).
'a non-starter' means to remove the first letter (the letter which starts the word is ignored).
'witch' with its first letter removed is 'ITCH'.
'in' is the link.
(Other definitions for itch that I've seen before include "It must be scratched" , "Hunger, longing" , "Burning" , "longing for something" , "Irritating feeling on the skin" .)