Royal person in car initially surrounded by media (8)
I believe the answer is:
princess
'royal person' is the definition.
'in car initially surrounded by media' is the wordplay.
'initially' suggests taking the first letters.
'surrounded by' means one lot of letters goes inside another.
'media' becomes 'press' (I've seen this before).
The first letter of 'car' is 'c'.
'in'+'c'='inc'
'inc' going within 'press' is 'PRINCESS'.
(Other definitions for princess that I've seen before include "Sovereign's daughter" , "VIP" , "Royal title" , "Royal daughter" , "Daughter of a sovereign" .)