Carriage from public house to a public school (7)
I believe the answer is:
phaeton
'carriage' is the definition.
(type of horse-drawn carriage)
'public house to a public school' is the wordplay.
'public house' becomes 'ph' (abbreviation).
'public school' becomes 'eton' (Eton College).
'ph'+'a'+'eton'='PHAETON'
'from' acts as a link.
(Other definitions for phaeton that I've seen before include "4-wheeled carriage" , "transport" , "Coach" .)