Assert to be right on the way (4)
I believe the answer is:
aver
'assert' is the definition.
(averring is a kind of asserting)
'right on the way' is the wordplay.
'right' becomes 'r' (common abbreviation).
'on' means one lot of letters go next to another.
'the way' becomes 'ave' (abbreviation for avenue, a type of 'way').
'r' put after 'ave' is 'AVER'.
'to be' is the link.
(Other definitions for aver that I've seen before include "Prove to be true" , "Declare or insist" , "protest" , "Profess" , "Say firmly, rave if necessary" .)