Long dog with no head (4)
I believe the answer is:
itch
'long' is the definition.
(to long or itch for something is to desire it)
'dog with no head' is the wordplay.
'dog' becomes 'bitch' (I've seen this before).
'with no head' means to remove the first letter.
'bitch' with its initial letter taken away is 'ITCH'.
(Other definitions for itch that I've seen before include "Skin sensation; desire" , "Seven-yearly desire" , "are dying" , "Constant desire" , "that's irritating" .)