You can start by looking on fanfiction websites like FanFiction.net or Archive of Our Own (AO3). They have a large collection of Naruto - related fanfictions, and you're likely to find some 'kakashi hug naruto' ones among them.
There are several 'kakashi hug naruto fanfiction' out there. For example, in some stories, the hug occurs during Naruto's training when he's frustrated and about to give up. Kakashi, seeing Naruto's struggle, gives him a warm hug. This hug not only comforts Naruto but also makes him realize that he's not alone in his journey. It symbolizes the mentor - mentee relationship they have. The fanfiction often explores the different emotions and the significance of this simple yet powerful gesture.
Well, there could be. Fanfiction writers love to explore unique and unexpected relationships. Since Naruto and Kakashi are both popular characters in the Naruto universe, some writers might come up with such a plot for the sake of creativity and exploring different storylines. But it's not a part of the official Naruto story.
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.
Another reason could be related to Kakashi's past actions. If there are some secrets or things Kakashi did in the past that come to light and Naruto doesn't like them, he could snap at Kakashi. Naruto has a strong sense of justice and loyalty, so if he feels betrayed or misled by Kakashi, his reaction could be to snap.
Kakashi might be initially surprised. But then, he would probably use it as a teaching moment for Naruto. He could say something like 'Naruto, stealing is not the way of a ninja.' And then give him a punishment like extra training.
One possible reason could be that Kakashi is preoccupied with other important matters in the village. Maybe he has some top - secret missions or tasks assigned by the Hokage that demand his full attention, leaving him with little time to focus on Naruto.
One reason could be that he sees Naruto as a younger version of himself in some ways. Both have had their share of loneliness and challenges growing up. Kakashi may want to prevent Naruto from going through the same pain he did. Another reason is Naruto's importance to the village. As the jinchuriki of the Nine - Tails, Naruto is a key asset. Kakashi, being a loyal ninja, wants to protect the village's future which is tied to Naruto.
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.
One memorable moment could be when Kakashi finally reveals a powerful new jutsu to Naruto that completely changes the way he approaches battles. Another might be when Kakashi saves Naruto from a near - fatal attack during a tough training session.