Drive past public area with king cheered heartily (8)
I believe the answer is:
overtake
'drive past' is the definition.
(to overtake in a car)
'public area with king cheered heartily' is the wordplay.
'public' becomes 'overt' (associated in meaning).
'area' becomes 'a' (maths abbreviation).
'with' says to put letters next to each other.
'king' becomes 'K' (abbreviation for king in chess).
'heartily' means to look at the middle letters.
The middle of 'cheered' is 'e'.
'overt'+'a'+'k'+'e'='OVERTAKE'
(Other definitions for overtake that I've seen before include "Pass the car in front" , "Catch unprepared" , "Leave behind" , "Catch up with and pass" , "Go past another person in a race" .)