New driver in Connecticut used half the medicine (7)
I believe the answer is:
linctus
'the medicine' is the definition.
(thesaurus)
'new driver in connecticut used half' is the wordplay.
'new driver' becomes 'l'.
'connecticut' becomes 'ct'.
'half' means to take half the letters.
'used' with half the letters taken is 'us'.
'l'+'in'+'ct'+'us'='LINCTUS'
(Other definitions for linctus that I've seen before include "The prescription" , "Cough remedy" , "Special formulation" , "Thick cough mixture" , "Soothing drink" .)