Prudish party leader is on edge (4)
I believe the answer is:
prim
'prudish' is the definition.
(I know that prudish can be written as prim)
'party leader is on edge' is the wordplay.
'party leader' becomes 'p' (leader of the word 'party').
'is on' is a charade indicator (letters next to each other) (in a down clue, letters appear on others).
'edge' becomes 'rim' (rim is a kind of edge).
'p'+'rim'='PRIM'
(Other definitions for prim that I've seen before include "Old-maidish" , "Proper or prissy" , "Prudish, proper" , "Affectedly formal" , "Straight-laced" .)