Escort and look after carriage heading for Oxford (6)
I believe the answer is:
gigolo
'escort' is the definition.
(term for a male escort)
'look after carriage heading for oxford' is the wordplay.
'look' becomes 'lo' (archaic exclamation meaning 'look!').
'after' is a charade indicator (letters next to each other).
'carriage' becomes 'gig' (gig is a kind of carriage).
'heading for' suggests taking the first letters.
The first letter of 'oxford' is 'o'.
'gig'+'o'='gigo'
'lo' put after 'gigo' is 'GIGOLO'.
'and' is the link.
(Other definitions for gigolo that I've seen before include "Paid male lover" , "Lounge lizard" , "Professional male escort" , "Male lover for hire" , "dancing partner" .)