Mark 1:44  . . . go show yourself to the priest, and offer for your cleansing the things which Moses commanded . . .

Every Jew knew that sacrifices could only take place in the Jerusalem temple (this had the effect of ensuring consistency) and, as Jesus said, they were obligated to follow to the letter the requirements prescribed by Moses as preserved in the Jewish Scriptures.

The sacrifices Jesus insisted on are detailed in the Scriptures cited below. You will see that the requirements varied according to the leper’s financial situation but the minimum involved the death of four creatures and that the priest [or any male in his family] was to eat at least two of them (and probably the one mentioned in verse 3 below). At least one of those eaten had to be a lamb.

The following is a little complex, so for those in a hurry, I highlight key parts and because it is long I include verse numbers.

Leviticus 14:2 “This shall be the law of the leper in the day of his cleansing. He shall be brought to the priest,
3 and the priest shall go out of the camp. The priest shall examine him, and behold, if the plague of leprosy is healed in the leper [known as leprosy in ancient times, though not necessarily what is called leprosy in modern times],
4 then the priest shall command them to take for him who is to be cleansed two living clean birds, and cedar wood, and scarlet, and hyssop.
5 The priest shall command them to kill one of the birds in an earthen vessel over running water.
6 As for the living bird, he shall take it, and the cedar wood, and the scarlet, and the hyssop, and shall dip them and the living bird in the blood of the bird that was killed over the running water.
7 He shall sprinkle on him who is to be cleansed from the leprosy seven times, and shall pronounce him clean, and shall let the living bird go into the open field.
8 “He who is to be cleansed shall wash his clothes, and shave off all his hair, and bathe himself in water; and he shall be clean. After that he shall come into the camp, but shall dwell outside his tent seven days.
9 It shall be on the seventh day, that he shall shave all his hair off his head and his beard and his eyebrows, even all his hair he shall shave off. He shall wash his clothes, and he shall bathe his body in water, then he shall be clean.
10 “On the eighth day [ie after the death of the above mentioned bird] he shall take two male lambs without defect, and one ewe lamb a year old without defect, and three tenths of an ephah of fine flour for a meal offering, mingled with oil, and one log of oil.
11 The priest who cleanses him shall set the man who is to be cleansed, and those things, before the Lord, at the door of the Tent of Meeting.
12 “The priest shall take one of the male lambs, and offer him for a trespass offering, with the log of oil, and wave them for a wave offering before the Lord.
13 He shall kill the male lamb in the place where they kill the sin offering and the burnt offering, in the place of the sanctuary; for as the sin offering is the priest’s, so is the trespass offering. It is most holy.
14 The priest shall take some of the blood of the trespass offering, and the priest shall put it on the tip of the right ear of him who is to be cleansed, and on the thumb of his right hand, and on the big toe of his right foot.
15 The priest shall take some of the log of oil, and pour it into the palm of his own left hand.
16 The priest shall dip his right finger in the oil that is in his left hand, and shall sprinkle some of the oil with his finger seven times before the Lord.
17 The priest shall put some of the rest of the oil that is in his hand on the tip of the right ear of him who is to be cleansed, and on the thumb of his right hand, and on the big toe of his right foot, upon the blood of the trespass offering.
18 The rest of the oil that is in the priest’s hand he shall put on the head of him who is to be cleansed, and the priest shall make atonement for him before the Lord.
19 “The priest shall offer the sin offering [a second lamb] sin offering, and make atonement for him who is to be cleansed because of his uncleanness: and afterward he shall kill the burnt offering; [a third lamb]
20 and the priest shall offer the burnt offering and the meal offering on the altar. The priest shall make atonement for him, and he shall be clean.
21 “If he is poor, and can’t afford so much, then he shall take one male lamb for a trespass offering to be waved, to make atonement for him, and one tenth of an ephah of fine flour mingled with oil for a meal offering, and a log of oil;
22 and two turtledoves, or two young pigeons, such as he is able to afford; and the one shall be a sin offering, and the other a burnt offering.
23 “On the eighth day he shall bring them for his cleansing to the priest, to the door of the Tent of Meeting, before the Lord.
24 The priest shall take the lamb of the trespass offering, and the log of oil, and the priest shall wave them for a wave offering before the Lord.
25 He shall kill the lamb of the trespass offering . The priest shall take some of the blood of the trespass offering and put it on the tip of the right ear of him who is to be cleansed, and on the thumb of his right hand, and on the big toe of his right foot.
26 The priest shall pour some of the oil into the palm of his own left hand;
27 and the priest shall sprinkle with his right finger some of the oil that is in his left hand seven times before the Lord.
28 Then the priest shall put some of the oil that is in his hand on the tip of the right ear of him who is to be cleansed, and on the thumb of his right hand, and on the big toe of his right foot, on the place of the blood of the trespass offering.
29 The rest of the oil that is in the priest’s hand he shall put on the head of him who is to be cleansed, to make atonement for him before the Lord.
30 He shall offer one of the turtledoves, or of the young pigeons, such as he is able to afford,
31 even such as he is able to afford, the one for a sin offering, and the other for a burnt offering, with the meal offering. The priest shall make atonement for him who is to be cleansed before the Lord.”
32 This is the law for him in whom is the plague of leprosy, who is not able to afford the sacrifice for his cleansing.
(Emphasis mine.)

Burnt offerings were rendered inedible by the flesh being fully consumed by fire but this did not apply to sin offerings and guilt offerings. As indicated in verse 13 (above) the sin offering and the guilt offering “belongs to the priest”. The following proves that this means they were to be eaten.

Leviticus 6:25-26, 29 Speak to Aaron and to his sons, saying, ‘This is the law of the sin offering: in the place where the burnt offering is killed, the sin offering shall be killed before the Lord. It is most holy. The priest who offers it for sin shall eat it. It shall be eaten in a holy place, in the court of the Tent of Meeting. . . . Every male among the priests shall eat of it: it is most holy.

Leviticus 7:5-6  . . . It is a trespass offering. Every male among the priests may eat of it. It shall be eaten in a holy place. It is most holy.

Leviticus 10:16-18 Moses diligently inquired about the goat of the sin offering, and, behold, it was burned: and he was angry with Eleazar and with Ithamar, the sons of Aaron who were left, saying, “Why haven’t you eaten the sin offering in the place of the sanctuary, since it is most holy, and he has given it you to bear the iniquity of the congregation, to make atonement for them before the Lord? Behold, its blood was not brought into the inner part of the sanctuary: you certainly should have eaten it in the sanctuary, as I commanded.”

Numbers 18:8-10 The Lord spoke to Aaron, “Behold, I myself have given you the command of my wave offerings, even all the holy things of the children of Israel. I have given them to you by reason of the anointing, and to your sons, as a portion forever. This shall be yours of the most holy things from the fire: every offering of theirs, even every meal offering of theirs, and every sin offering of theirs, and every trespass offering of theirs, which they shall render to me, shall be most holy for you and for your sons. You shall eat of it like the most holy things. Every male shall eat of it. It shall be holy to you.

(Emphasis mine.)