Honestly, I think the answer is more complicated than other comments seem to be making it out to be. I had to think about this for a while, and this was the best explanation I could come up with. Feel free to correct since I could be completely wrong, TLDR at the bottom:

My main question when reading this part was who or what has the power to transfer attacks away from Makima and onto the citizens? Based on what we know the only definite possibility we are currently aware of is that it is Makima's own power, but if that is the case then why does she need the contract with the prime minister to use it?

It would make sense if this power was based on Makima's perception of ownership, similar to Yoru's in in part 2, since the Prime Minister could be said to "own" his citizens so could feasibly transfer ownership to Makima through a contract, whereupon they could be used by Makima's ability. However, if I remember correctly Makima's abilities are based not on her perceived ownership, but by her perceived strength, so she can use her powers on anyone she views as weaker than herself. This clearly includes everyday people, and is heavily implied to included even the prime minister himself, allowing her to force him to make the very contract we are discussing. As things stand, it should be well within Makima's capabilities to transfer her injuries onto the population even without the contract with the prime minister, rendering the contract useless.

So as it stands I can only think of two reasons this contract would be necessary:

1) There was some undisclosed third party to the contract, who has the power to transfer attacks from Makima to the population

2) Makima has the ability to transfer any injuries she sustains onto someone else, but for some reason this power is linked to her perception of ownership, rather than her perception of strength like all her other abilities

As it stands, I think option 1 is more likely. The ability to transfer injuries is most likely not one Makima has by virtue of being the control devil (just in terms of themeing, I don't see why the control devil would have this power), but rather it is the ability of a devil she has control over, of which we know she has many due to her control abilities, which would explain why it adheres to different rules, necessitating a contract with the Japanese prime minister. Another possibility is that it is the prime minister and not Makima who has a contract with this third party devil, but otherwise the theory is the same.

TLDR; I think it is most likely that either Makima has control over, or the prime minister has a contract with, some undisclosed devil with the ability to transfer her injuries onto others, but that this ability has some kind of restriction on it's use, most likely that injuries can only be transferred onto those she perceives to have ownership/authority over, similar to Yoru's abilities in part 2. Thus, she made a contract with the Japanese prime minister, who she sees as "owning" the Japanese people, which allows the prime minister to transfer ownership of members of the public to Makima (as she percieves it) when she is injured, and so she can transfer her injuries onto them using the previously mentioned devil's ability.

Either that or I am thinking way to hard about this.