Prince Henry follows car, shaking fist (8)
I believe the answer is:
archduke
'prince' is the definition.
(archduke is a kind of prince)
'henry follows car shaking fist' is the wordplay.
'henry' becomes 'h' ('H' can be a synonym of 'henry').
'follows' says to put letters next to each other.
'shaking' is an anagram indicator.
'fist' becomes 'duke' (I've seen this before).
'car' anagrammed gives 'arc'.
'h' after 'arc' is 'arch'.
'arch'+'duke'='ARCHDUKE'
(Other definitions for archduke that I've seen before include "noted trio" , "Top nobleman" , "Title of Franz Ferdinand, shot in Sarajevo in 1914" , "Emperor's son" , "Austrian noble" .)