In such a fanfiction, the plot might involve Naruto turning 'dark' due to feelings of abandonment. He could become bitter and start to question the values Jiraiya taught him. Maybe he'd start using his powers in more ruthless ways, seeing the world as having turned its back on him just as Jiraiya did. He might form alliances with darker characters in the Naruto universe, believing they are the only ones who understand his pain.
Naruto's character would become more cynical. He'd lose his usual optimism and be more distrustful of others. His sense of justice might be warped, as he now feels that he has been wronged. He could start to see power as the only thing that matters, rather than the bonds he once cherished.
Sakura might be worried about Naruto. She could try to comfort him and encourage him not to give up. She might also be angry at Jiraiya for abandoning Naruto, seeing how much Naruto has been through already. Kakashi, on the other hand, would probably step in more as a mentor. He has his own way of teaching, and he'd want to make sure Naruto doesn't go astray.
Well, it could make Naruto more distrustful. He's already had a tough life, and being abandoned by Jiraiya, who was like a father figure, would likely shatter his sense of security. This might lead him to become more self - reliant and closed off initially. But in the long run, it could also fuel his determination to prove himself even more, without relying on Jiraiya's guidance.
Naruto may also develop a darker side to his personality. The abandonment could make him feel betrayed and angry. He might struggle with controlling his emotions, especially his anger. However, this could also lead to him learning how to channel that anger into power in a more controlled manner, like he did with the Nine - Tails' chakra, but in a more refined way.
One possible plotline could be Naruto feeling extremely lost and having to find his own way to become stronger without their guidance. He might start to question his own worth and abilities, but then he could meet new allies who help him realize his true potential on his own. For example, he could encounter some other outcast ninja who have their own unique skills and together they form a new kind of ninja team that challenges the traditional ways of the ninja world.
Jiraiya might comfort Naruto by sharing his own experiences. He could tell Naruto about the times he faced difficulties and how he overcame them. For example, he might talk about his training with the toads and how he grew stronger through perseverance.
Perhaps Naruto starts to question his own identity. Since he was abandoned by his family, he could be on a journey to discover his true heritage and in the process, his dark side grows as he faces many hardships and betrayals. He may then use his new understanding to either wreak havoc or find a way to forgive and move forward.
Well, it could be that Dark Naruto, after being abandoned, is found by an outcast group. They take him in and teach him their own values which are different from the village's. He grows up with a different perspective and starts to question the whole idea of the ninja world. Eventually, he might try to create his own 'utopia' in his mind, which could involve overthrowing the current system. But at the same time, he still has that internal conflict of his Uzumaki heritage and what Kushina's abandonment means.
Perhaps in this fanfiction, Jiraiya discovers that Naruto is going to be the cause of a major catastrophe in the future. He might have seen a prophecy or some sort of sign that convinces him Naruto has to die to prevent this. It could also be that Jiraiya has been misled by false information about Naruto, which makes him think that Naruto has become an evil force that needs to be eliminated.