Paid escort in carriage, alone, scratching head (6)
I believe the answer is:
gigolo
'paid escort' is the definition.
(I've seen this before)
'carriage alone scratching head' is the wordplay.
'carriage' becomes 'gig' (gig is a kind of carriage).
'alone' becomes 'solo' (as in a musical solo).
'scratching head' means to remove the first letter (the first letter or head is scratched out).
'solo' with its first letter removed is 'olo'.
'gig'+'olo'='GIGOLO'
'in' is the link.
(Other definitions for gigolo that I've seen before include "Male lover for hire" , "Professional male escort" , "Man living off an older woman" , "Paid male lover" , "Hired male escort" .)