Public argument about start of his deposition (9)
I believe the answer is:
overthrow
'deposition' is the definition.
(thesaurus)
'public argument about start of his' is the wordplay.
'public' becomes 'overt' ('overt' is associated in meaning with 'public'**).
'argument' becomes 'row' (I've seen this before).
'about' says to put letters next to each other.
'start of' indicates taking the first letters.
The initial letter of 'his' is 'h'.
'row' after 'h' is 'hrow'.
'overt'+'hrow'='OVERTHROW'
(Other definitions for overthrow that I've seen before include "Trevor who? (anag)" , "Defeat completely" , "Bring extra" , "Bring down a ruler" , "Unexpected run" .)