The desire of a Billy No-Mates? (5,4)
I believe the answer is:
nanny goat
'no-mates?' is the definition.
I can't tell whether this definition defines the answer.
'the desire of a billy' is the wordplay.
'the desire of' becomes 'nanny' (I can't justify this - if you can you should give a lot more credence to this answer).
'a billy' becomes 'goat'.
'nanny'+'goat'='NANNY-GOAT'
(Other definitions for nanny goat that I've seen before include "Billy's partner?" , "Butter" , "Female farm animal" , "milk producer" .)