Naruto's relationship with the other Jinchuriki would likely change. If Naruto hates Yondaime, he might see the system of creating Jinchuriki in a more negative light. He could try to convince the other Jinchuriki that they have been wronged by the villages and their leaders like Yondaime. This could lead to either a united front among the Jinchuriki against the villages or some internal conflict among them depending on their individual beliefs.
Well, in the original, Yondaime was Naruto's father and a hero. But in this fanfiction, if Naruto hates him, it completely flips the relationship. It makes Yondaime more of a villain - like figure in Naruto's eyes.
Perhaps Naruto blames Yondaime for his mother's death. In the original story, Kushina died while giving birth to Naruto, and if in this fanfiction there are some hidden motives or actions of Yondaime related to that event, Naruto could hold a grudge. Like if he found out that Yondaime could have saved Kushina but chose not to for some greater good that Naruto doesn't understand or accept.
His relationship with Minato would be very different. Instead of a father - son bond, it would be more of a brother - in - law type of relationship. Minato might see Naruto as an equal in a different way, and they could collaborate more as peers. Also, Naruto's relationship with Sasuke would have a new angle. Sasuke might be more envious of Naruto's strong Uzumaki bloodline, which could intensify their rivalry.
For Naruto and his friends, his new connection to the Dragon Sage power from Kushina could make some of his friends a bit jealous at first. However, as they see how he uses it for the good of the village, they would come to accept it. His relationship with Sasuke could also be affected. Sasuke might be intrigued by this new power and see it as a way to also become stronger, which could lead to some friendly competition between them. With Kushina, Naruto would share a deeper emotional connection as they explore their shared power heritage together.
If Kushina was alive in a fanfiction, Naruto's relationship with his father, Minato, would be different. They could all be a proper family, and Naruto would grow up with both his parents. His relationship with his friends might also change as he would have a more stable home life. For example, he might be more confident around Sakura and Sasuke, and his bond with them could be less about seeking acceptance due to his lack of family.
If they are revived, Naruto's relationship with his parents would be much more complex. He has grown up without them, and now he has to adjust to having them around. Minato and Kushina would also have to get used to the fact that Naruto is no longer the little boy they left behind. Their relationship would be filled with a mix of catching up and learning about each other all over again.
One possible reason could be a misinformation plot. Maybe in the fanfiction, Naruto has been led to believe that Yondaime had some dark secrets or did something bad to his family that he discovers later in the story.
In such a fanfiction, the relationship between Naruto and Kushina would be one of mother - son love and mutual respect. Kushina would be protective of Naruto and proud of the man he has become. Naruto would look up to Kushina and value her wisdom. With Hinata, Naruto's relationship would be that of a loving couple. Hinata would support Naruto in all his endeavors. When it comes to Kushina and Hinata, Kushina would likely see Hinata as a worthy partner for Naruto and Hinata would respect Kushina for being Naruto's mother and a strong kunoichi.
If Naruto becomes Anko's apprentice, his relationship with Kakashi would be affected. Kakashi might be a bit surprised at first but would also be interested to see how Naruto grows under Anko's guidance. Naruto's friends in Team 7 would also have different reactions. Sakura could be curious and a bit envious of the new skills Naruto is learning. As for Sasuke, he might view Naruto as a threat in a different way now. Naruto's new skills might make Sasuke more determined to surpass him. Also, Naruto's relationship with the other villagers could change. Some might be a bit wary of his new - found skills, especially if they are related to Anko's more unorthodox and perhaps a bit dangerous techniques. But overall, Naruto's apprenticeship with Anko would cause a ripple effect in his relationships throughout the village.
Naruto would become more confident in his abilities. With Kushina's training, he would have a better understanding of his Uzumaki heritage and the power that comes with it. This new - found confidence would show in his battles and his interactions with others.