One possible reason could be some sort of dark influence or curse that makes her lose her maternal instincts. Maybe a powerful enemy targets her and in order to protect Naruto from further danger related to her, she feels forced to abandon him. Another reason might be that she is made to believe false information about Naruto, such as that he will bring great disaster to the village in the future.
In such a fanfiction scenario, the village of Konoha would be affected too. Minato is a respected figure. His abandonment could cause a scandal. Naruto might not have the same level of protection from the village elders. This could lead to different power dynamics within the village. Maybe other ninja clans would try to take advantage of Naruto's situation. Also, Kushina's family might react in various ways, some might try to find Minato and some might focus on taking care of Kushina and Naruto.
He may develop trust issues. Being abandoned by his mother could make him more cautious about getting close to people. However, this could also make him more determined to prove himself. He might strive to be the best ninja in the village to show that he is worthy of love and acceptance, despite being abandoned. For example, he could push himself extra hard during training.
Naruto might initially be in shock. He has always counted on Kushina's love, and this sudden abandonment would be a huge blow. He could become withdrawn and quiet, not understanding why his mother would do such a thing.
In the fanfiction, it could be that Kushina is pressured by external forces. For example, there might be a powerful group or individual who forces her to abandon Naruto for his sister. This could be part of a larger political or power - play scheme. Also, it's possible that Kushina has a moment of extreme emotional confusion. She might see something in Naruto's sister that she once desired for herself when she was younger, and in a moment of weakness, she decides to focus on the sister instead of Naruto.
It could be a matter of prophecy. Minato might have received some sort of prophecy that staying with Naruto and Kushina would lead to the destruction of the village. So, in a misguided attempt to save Konoha, he abandons them. Or, there could be a power struggle within the village that makes Minato believe that leaving Naruto and Kushina is the only way to keep the peace. However, these are all just speculations within the context of fanfiction.
It would have a profound impact. Naruto's journey in the original was driven by his desire to be recognized and loved. If Kushina abandons him, he might take on a more survival - based mindset. He'd have to fend for himself from a very young age. This could make him more street - smart but also more bitter. His ninja way might change to something more self - serving as he feels he has only himself to rely on.
Another aspect could be that the fanfiction delves into how Kushina's presence affects Naruto's relationships with others. For instance, his relationship with Sasuke could be different. With a strong maternal figure in his life, Naruto might be more stable emotionally and be able to approach Sasuke in a more understanding way, which could potentially change the whole dynamic of their rivalry and friendship.
Well, it could go in many directions. For example, his family might start to search for him, which could lead to a series of adventures where they encounter new characters and face various challenges. His wife, if he has one in the story, might become stronger and more independent in his absence, vowing to bring him back.
Naruto would be devastated. He sees Kakashi as a mentor and father figure. He might initially go into denial, not believing that Kakashi would actually abandon him.