It could be due to a misunderstanding. Kakashi might think that Naruto is getting involved with something dangerous or has some potential that he can't handle. So, in his warped way of thinking, he abandons Naruto to protect him from himself or from some unknown danger.
Naruto could initially feel a great sense of loss and sadness. But as time goes on, he might start to look for new mentors or friends who can support him. For example, he could turn to Iruka-sensei more often or form closer bonds with his fellow classmates like Shikamaru or Choji. He'd rely on his inner strength and the relationships he already has to get through this tough time.
One possible main plotline could be Naruto growing up with Kakashi's unique training methods. Since Kakashi is known for his strict and strategic training, Naruto might develop different skills compared to the original series. Another plotline could be how Naruto deals with the abandonment issues from his past while having Kakashi as a father figure. He could struggle with trust issues at first but gradually open up as Kakashi shows him unconditional love and support.
Naruto's character would likely be more reserved in some ways. Kakashi is a rather stoic and composed person, and his influence might make Naruto less brash. Instead of always charging headfirst into situations, he would think more. For example, in battles, he might analyze the situation like Kakashi does before acting.
His ninja skills would probably develop in a different way. Without Kakashi's guidance, he'd have to figure things out on his own. This might lead to some unorthodox training methods and a unique fighting style. For example, he might focus more on his natural instincts and less on the traditional teachings he received from Kakashi. He could also form new alliances with characters who are also outsiders or misfits in the ninja world, creating a new support system for himself.
One possible reason could be a mission gone wrong. Maybe Kakashi thought Naruto was too much of a liability during a particularly dangerous mission and, out of a sense of harsh pragmatism, decided to abandon him to ensure the success of the mission and the safety of others. However, this would be very out of character for Kakashi as he has always been protective of his students.
Well, Naruto's character would be different in many ways. Firstly, he'd have a stronger sense of discipline. Kakashi would instill in him the importance of following rules and regulations in the shinobi world. His view of friendship might also be different. While he'd still value his friends, he might be more guarded initially, as Kakashi might teach him to be cautious. His relationship with the Hokage and the village elders would also be distinct. He might have more respect for their authority due to Kakashi's influence, but at the same time, he could be more critical of their decisions if he felt they were not in the best interest of the village, based on the knowledge and values Kakashi has imparted to him.
In this fanfiction, a great plot development could be the impact on the story's antagonists. Since Naruto is being raised differently, his growth might be noticed earlier by the likes of Orochimaru or the Akatsuki. This could lead to earlier and more intense confrontations. Moreover, Kakashi's influence on Naruto's moral code could make Naruto more of a grey - area character. He might be more willing to make tough decisions and sacrifices for the greater good, which would completely change the course of many story arcs.
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.