Often, Naruto is paired with Kakashi in such fanfictions. Their bond from the original series can be explored in new ways in the context of Kakashi's reincarnation. For example, Naruto might be the one to discover Kakashi's reincarnated form and help him adjust.
In some fanfictions, Kakashi might be paired with characters from other villages. For instance, Gaara. Maybe in his reincarnated state, Kakashi ends up in Sand Village and forms an unexpected alliance with Gaara as they face a new common threat.
Often, Gai is paired with Kakashi in Hurt Kakashi fanfiction. Gai, with his unwavering belief in friendship and his eternal rivalry with Kakashi, can be a great character to interact with a hurt Kakashi. He might try to pull Kakashi out of his slump through their friendly competition. Also, Rin is sometimes included. Since she has a significant past with Kakashi, her presence in the fanfiction can either be a source of comfort or more pain, depending on how the story is written.
Naruto can also be paired with Dark Kakashi. Naruto's unwavering belief in the good of people could contrast sharply with Kakashi's darker self. It might lead to situations where Naruto tries to bring back the old, more positive Kakashi, and through their interactions, both characters grow and learn more about themselves.
Naruto is often a common character. Since Naruto has a strong bond with Kakashi, he might be involved in Kakashi's time - traveling adventures. Sasuke could also be there. Their relationship with Kakashi in the original story makes it likely for them to show up. And of course, Guy would be a great addition. His friendly rivalry with Kakashi would add some humor and action to the time - travel scenarios.
Cheelai and Lemo from the Broly movie might also be paired with the reincarnated Broly. They could act as his guides or friends in the new world he finds himself in, helping him navigate the complex politics and power dynamics of the Dragon Ball universe.
Sakura could also be present. She might be observing the training and learning some things herself, or she could be helping Kakashi in some way, like providing medical support during intense training exercises.
Sasuke could be a popular character. He might be involved in the story in a way that either helps or hinders the situation where Naruto hurts Kakashi. For example, if Sasuke tries to stop Naruto from doing something stupid that would hurt Kakashi.
I think Rin would be an interesting character to pair with Kakashi in time travel fanfictions. Since she was an important part of Kakashi's past, their relationship could be explored further. Maybe Kakashi travels back to save her from her tragic fate, and they end up having a different kind of relationship as a result.
Perhaps a plotline where Kakashi is reincarnated in a world where the Akatsuki has achieved their goals. He would then be in a situation where he has to figure out how to oppose them or find a way to coexist in this new, darker ninja world.
Kakashi is often a favorite. His cool and mysterious demeanor, along with his powerful jutsu, make him a central and exciting character in fanfiction. He has a lot of room for backstory exploration.
Iruka could be included. He was Naruto's first teacher and has his own connection to Kakashi. His presence can bring out the different teaching styles between him and Kakashi. Another good character is Jiraiya. His relationship with Naruto as his godfather and his connection to Kakashi through being comrades can add some interesting storylines.