Absolutely nothing! The reason the three pesukim are added at the beginning is so that we don't have to add at the end.
To explain, consider Shulchan Aruch, Orach Chaim 284:
מפטירין בנביא מענינה של פרשה ואין פוחתין מכ"א פסוקים אלא אם כן סליק ענינא בבציר מהכי כגון עולותיכם ספו על זבחיכם.
"We read the Haftarah from the Navi from the subject matter of the parasha. And we do not read less than 21 verses unless the topic has finished with less than that, such as [the haftarah of Tzav], עולותיכם ספו על זבחיכם [which starts in Yirmeyahu 7:21 and extends to 7:34, then 8:1-3, and 9:22-23, which is less than 21 verses].
Now the haftarah of Tazria, as mentioned in the question, runs from Kings Ⅱ 4:42-5:19. That is, three (seemingly) irrelevant pesukim from the end of the previous perek, about bringing loaves, and then 19 verses from perek 5.
If there were only the 19 verses from perek 5, that would not yield the requisite 21 pesukim. With the three additional pesukim, the sum total is 22.
Why not continue on in the story at the end? Because we end at a good place, with the kiddush Hashem. The navi had refused payment, and Naaman leaves, very impressed.
It gets worse from then on... Looking at the Haftarah in English, here is the continuation, pasuk 20 and on:
"20: And Gehazi, the servant of Elisha the man of God, thought, "Here my master has stopped Naaman, this Aramean, from giving, by not taking from his hand what he brought. As the Lord lives, I will run after him and take something from him." 21: And Gehazi chased after Naaman; and Naaman saw him running after him, and he leaned over off the chariot toward him, and said, "Is all well?" 22: And he said, "All is well. My master sent me, saying, "Here, just now two youths have come to me from Mt. Ephraim, of the disciples of the prophets. Please give them a talent of silver and two suits of clothing. 23: And Naaman said, "Please take two talents. And he urged him and he tied two talents of silver in two pockets, and two suits of clothing. And he gave his two servants, and they carried them before him. 24: And he came to a secret place, and he took [them] from their hands, and he deposited them in the house. And he dismissed the men, and they went away. 25: And he came and stood before his master, and Elisha said to him, "Where are you coming from, Gehazi?" And he said, "Your servant has gone neither here nor there." 26: And he said to him, "Did my heart not go when a man turned around off his chariot toward you? Is it time to take the silver, and to buy clothing and olive trees and vineyards and sheep and cattle and slaves and maidservants? 27: Now Naaman's zaraath shall cling to you and to your children forever." And he went away from before him, stricken with zaraath, [white] as snow."
Now, we cannot just end at pasuk # 19 and say the topic is done, because the story absolutely does continue. And so, instead, we pad at the beginning, so that there are 21 verses, and so that we can stop in the place we desire.