The first object of the predicate "require" is "the correction." Now, "the counteraction" is the second?
Plus, "in characters" = "in words?"
"Executive duties" = "the duties of the President (of the United States)"?
Context:
The recent demonstration of public sentiment inscribes on the list of
Executive duties,
in characters too legible to be overlooked, the task of reform, which will
require particularly the correction of those abuses that have brought the patronage of the Federal Government into conflict with the freedom of elections,
and the counteraction of those causes which have disturbed the rightful course of appointment and have placed or continued power in unfaithful or incompetent hands.
More:
http://avalon.law.yale.edu/19th_century/jackson1.asp