Well, Kakashi might start by spending more time with Naruto. He could train Naruto more intensively, like teaching him new jutsu. Maybe he also shares his own experiences to give Naruto some guidance and motivation.
In such a fanfic, Naruto's character development could be severely affected. He might become more withdrawn and lack the confidence he gains with Kakashi's proper guidance. Without Kakashi's attention, he may struggle to learn advanced ninja techniques as effectively, leading to a slower growth in his skills. This could also make him more vulnerable to the negative influences around him, like the villagers' scorn, and might even drive him towards darker paths in search of power and recognition.
In the fanfic, Kakashi might neglect Naruto's training due to a variety of reasons. He could be preoccupied with his own research or a secret mission that demands his full attention. There could also be a misunderstanding between him and Naruto. Maybe Naruto has done something that Kakashi misinterprets, causing him to distance himself from Naruto's training. Additionally, the fanfic could introduce new characters or plot elements that draw Kakashi's focus away from Naruto. For example, a new threat to the village that Kakashi feels he must handle alone, thus neglecting his duties as Naruto's trainer.
Maybe Kakashi is preoccupied with other important missions. He is a highly - skilled and often - needed ninja. There could be some top - secret missions that require his full attention, leaving him no time to train Naruto properly.
Naruto might also try to find alternative ways to gain strength. He could explore the forest around the village to find new training areas or discover new chakra - related phenomena. He has a curious nature, and this could lead him to stumble upon something that gives him an edge in his training, like a hidden hot spring that has special chakra - enhancing properties.
One common plot is Naruto seeking alternative mentors. Since Kakashi is neglecting him, he might look for other ninja in the village who can teach him. Another plot could be Naruto trying to prove himself to Kakashi by achieving great feats on his own. Some fanfics also show Naruto developing a closer bond with other outcasts in the village as he feels neglected by Kakashi, who is supposed to be his teacher.
In such a fanfiction, Naruto might become extremely powerful under Kakashi's training. He could learn new jutsus faster and gain better control over his chakra. Maybe he'd also develop a closer bond with Kakashi, different from his relationship with Iruka or other characters in the original story.
One common plotline is Naruto seeking out other mentors. Since Kakashi is neglecting him, he might look for someone like Jiraiya earlier than in the original story.
The neglect in the fanfic probably makes Naruto's growth more difficult. He might not have proper guidance, so his chakra control or combat skills may not develop as smoothly. This could also make his relationships with other characters more strained. For example, he might envy those who get more attention from Kakashi. Without proper training, he may struggle more in battles and his overall progress in becoming a stronger ninja would be hampered.
A plot development could be that Kakashi discovers the root cause of Naruto's neglect. It might be related to some hidden secrets in the village. Then Kakashi has to fight against those who are behind Naruto's neglect while still training Naruto. Another development could be that Naruto and Kakashi form a very strong bond. Naruto then uses the skills and confidence he gained from Kakashi to face his enemies and also to try and change the village's attitude towards him.