Well, for starters, Kakashi has to deal with Naruto's hyperactive and sometimes reckless behavior. Naruto often acts without thinking, which could put him in danger. Sasuke, with his complex emotions and the burden of his clan's massacre, is also a handful. Kakashi needs to make Sasuke understand that revenge is not the only way forward. Additionally, Kakashi has to ensure that both Naruto and Sasuke get along well with each other. There could be rivalry between them, and Kakashi has to manage that in a way that doesn't harm their relationship or their growth as ninjas.
One challenge could be Naruto's stubbornness. Naruto has his own ideas and ways of doing things, and getting him to listen and follow Kakashi's guidance might be tough. Another is dealing with Naruto's past traumas. Kakashi would need to help Naruto overcome his feelings of loneliness and being an outcast. Also, Kakashi has his own responsibilities and missions, so finding time to be a proper adoptive father could be a struggle.
Kakashi would become more nurturing. In the original, he was a teacher, but as an adoptive father, he'd have to be more affectionate. He might show his face more often to Naruto, as a sign of trust, and be more open about his emotions.
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It could make her more confident. With Kakashi's guidance, she would gain more skills and knowledge, which would boost her self - assurance in battles and in general ninja life.
One possible reason could be that Kakashi might be haunted by his past experiences. He has lost many comrades, and perhaps he's afraid that getting too close to Naruto and training him would put Naruto in danger as well. He might think that Naruto, being the Jinchuriki of the Nine - Tails, already has a target on his back, and training him would only make that target bigger.
Kakashi would probably be very calm at first. He's an experienced ninja who has seen a lot. He'd gently pat Naruto on the back, offering silent comfort. Then he might start to speak softly, asking Naruto what's wrong in a gentle voice.
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.