Fiery speech supporting extremely tetchy, oppressive boss (6)
I believe the answer is:
tyrant
'oppressive boss' is the definition.
(I've seen this in another clue)
'fiery speech supporting extremely tetchy' is the wordplay.
'fiery speech' becomes 'rant' (I've seen this in another clue).
'supporting' says to put letters next to each other (in a down clue, some letters go below others).
'extremely' suggests removing the centre (only the extremes of the word are used).
'tetchy' with its middle taken out is 'ty'.
'rant' put after 'ty' is 'TYRANT'.
(Other definitions for tyrant that I've seen before include "Autocrat or despot" , "Martinet" , "big bully" , "Totalitarian" , "Despot, dictator" .)