In the fanfiction, a possible backstory could be that Naruto, after achieving great power and peace in his own time, decided to use his abilities to explore the past. During this exploration, he accidentally or purposefully got involved in the lives of his ancestors. He might have fallen in love with a woman from the Uzumaki clan in the past and had children. These children then passed on their genes through generations until Kushina was born. It could also be that there was some sort of experiment by an ancient shinobi that went wrong or right (depending on how you look at it). This experiment somehow connected Naruto's future self to be Kushina's grandfather by altering the fabric of time and family relations in a very strange way.
Perhaps in this fanfiction, Naruto travels back in time. He goes to a time when the world was being formed, and through his unique chakra, he gives birth to the Nine - tails. This could be part of a grand plan to ensure balance in the future ninja world, and he is the only one with the power to do so.
Maybe Nagato had a relationship with an Uzumaki woman before he became fully involved in the Akatsuki. This woman could have given birth to Naruto but for some reason, Naruto was separated from Nagato at a young age. It could be due to the chaos and danger that Nagato was getting into with his plans for the Akatsuki.
It could be that the wolf is a reincarnation of an ancient protector of the ninja world. It was waiting for the right person to bond with, and Naruto, with his strong will and desire for peace, was the chosen one. So they became partners.
Maybe there was some sort of divine interference or prophecy. For instance, a prophecy foretold that a child of Artemis would be the key to saving the world from a great threat. So, the gods found a way to make it happen, despite Artemis' vow of chastity. It could also be that some dark magic forced Artemis into a situation where she had to bear a child, and that child was Percy.
One possible reason could be that Kushina has a very strong and intimidating personality. In the Naruto universe, she was known to be a tough kunoichi. Naruto, who might be used to more gentle or reserved people around him, could be scared of her boldness.
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.
One possible reason could be a mission. In many Naruto - related stories, missions often take characters away from their families. Naruto might have received a highly important and dangerous mission that requires him to be away from Kushina for a long time.
It could also be due to some external influence. Perhaps an enemy used some kind of mind - control or manipulation on Kushina. They made her think that Naruto was a threat, so she decided to disown him. In the Naruto universe, there are many powerful villains who could potentially do this kind of thing.
Maybe she's been possessed. In some fanfictions, characters can be taken over by dark forces. So Kushina could be under the control of some evil entity and that's why she hurts Naruto.
Maybe he had some bad experiences with weaklings who let him down in crucial moments. For example, a weak comrade could have failed to protect someone important, and that made Naruto start to dislike those he considered weak.