Said to have begun right away (6)
I believe the answer is:
stated
'said' is the definition.
('state' can be a synonym of 'say')
'begun right away' is the wordplay.
'begun' becomes 'started' ('start' can be a synonym of 'begin').
'right' becomes 'r' (common abbreviation).
'away' suggests deleting specific letters.
'started' with 'r' removed is 'STATED'.
'to have' is the link.
(Other definitions for stated that I've seen before include "Announced" , "Said firmly, averred" , "Asserted, affirmed" , "Close-fitting; drunk" , "Uttered, expressed in words" .)