Head of enquiry on free trip (4)
I believe the answer is:
ride
'trip' is the definition.
(thesaurus)
'head of enquiry on free' is the wordplay.
'head of' says to take the initial letters.
'on' is a charade indicator (letters next to each other).
'free' becomes 'rid' ('rid' can be a synonym of 'free'**).
The initial letter of 'enquiry' is 'e'.
'e' after 'rid' is 'RIDE'.
(Other definitions for ride that I've seen before include "Travel on (a bicycle)" , "Take a trip" , "Travel on the back of animal" , "Journey on horseback" , "Go by horse" .)