The Sin Offering tor a Ruler, or tribal chief or representative. The word is also used of the one chief of the nation in post-exilic writers when the succession of kings had come to an end. It would apply to Nehemiah, or perhaps to a foreign ruler like the Persian Bagoas, governor of Jerusalem in 402 B.C. The offering is a goat instead of a bullock, and its blood is only smeared on the horns of the altar, not sprinkled, and, as it would seem, by an ordinary priest, not the High Priest.
Leviticus 4:22-26
22 When a ruler hath sinned, and done somewhat through ignorance against any of the commandments of the LORD his God concerning things which should not be done, and is guilty;
23 Or if his sin, wherein he hath sinned, come to his knowledge; he shall bring his offering, a kid of the goats, a male without blemish:
24 And he shall lay his hand upon the head of the goat, and kill it in the place where they kill the burnt offering before the LORD: it is a sin offering.
25 And the priest shall take of the blood of the sin offering with his finger, and put it upon the horns of the altar of burnt offering, and shall pour out his blood at the bottom of the altar of burnt offering.
26 And he shall burn all his fat upon the altar, as the fat of the sacrifice of peace offerings: and the priest shall make an atonement for him as concerning his sin, and it shall be forgiven him.