Like speech praising old archbishop, a right-winger (9)
I believe the answer is:
laudatory
'like speech praising' is the definition.
(like a speech lauding someone)
'old archbishop a right-winger' is the wordplay.
'old archbishop' becomes 'Laud' (William Laud, Archbishop of Canterbury 16331645).
'right-winger' becomes 'Tory' (Tory is an example).
'laud'+'a'+'tory'='LAUDATORY'
(Other definitions for laudatory that I've seen before include "Full of praise" , "puffing" , "Expressing praise and commendation" , "Eulogistic" .)