Naruto might be longing for a motherly figure in his life. He has always been alone and has seen how others have mothers. Tsunade, being a strong and caring woman in the village, could be someone he looks up to and wishes to have as a mother.
There are multiple possible reasons. Naruto has grown up without a mother, and he has seen how families function. Tsunade is not only a powerful shinobi but also someone who has a soft side. He may have noticed how she cares about the village and its people, and he wants that care directed towards him on a more personal level. He might also be tired of always being alone in a certain sense. He wants someone to scold him when he's wrong, comfort him when he's sad, and be proud of him like a mother would. Tsunade, with her status and character, seems like a suitable person for him to make this request to.
One reason could be that Naruto is seeking more emotional support. He has friends, but a mother is a different kind of relationship. Tsunade has shown kindness to him at times. Also, he may feel that having a mother - like Tsunade - would give him a sense of family that he has always craved. With his own parents gone, Tsunade represents stability and care that he wants in his life.
One possible reason could be extreme guilt. Maybe Naruto has done something he believes is unforgivable, like accidentally causing the death of a beloved friend while in a fight. He might think that death is the only way to atone for his mistake.
It might also be due to a sense of hopelessness. If he's faced with an enemy that seems truly unbeatable and he fears the destruction of the village or his friends, he could ask to be killed to prevent further harm. For example, if there's a powerful villain who has Naruto's friends hostage and the only way to save them is by Naruto sacrificing himself.
She might be initially shocked. Given her own past traumas and her relationship with Naruto, she could be hesitant at first. But then, considering Naruto's lonely situation, she might warm up to the idea and show a more maternal side towards him.
Perhaps he has been through some sort of mental and emotional breakdown. Maybe he has lost too many battles, seen too many friends die, and he feels that he is a burden or that his death would somehow bring peace to his troubled mind. It could also be related to some prophecy or dark secret he has discovered about himself that makes him think death is the best option for everyone.
One possible reason could be that Naruto feels he has become a threat to the village or his friends in some way. Maybe he has some sort of dark power within him that he can't control and he believes Sasuke is the only one strong enough to stop him. Another reason might be related to a prophecy or a greater threat that Naruto thinks his death can prevent, and he trusts Sasuke to carry out this difficult task.
Maybe Naruto discovers some dark secret about Tsunade's past that she has been trying to hide. For example, something related to her lost loved ones. Naruto, in his usual straightforward way, confronts Tsunade about it without realizing how much it would hurt her, and thus makes her cry.
One possible reason could be a misunderstanding. Maybe Tsunade received some false information about Naruto that made her think he was a threat to the village, so she took actions that ended up hurting him.
One possible reason could be a misunderstanding. Maybe some false information was spread about his mother, making Naruto believe she had abandoned him. Another reason could be related to the secrecy around his mother. If he grew up feeling alone and not knowing about her, he might develop resentment.
One reason could be a mission. Maybe he gets a top - secret mission that requires him to leave the village under Tsunade's leadership without her direct involvement. So he has to leave without much notice.
One possible reason could be misinformation. Maybe in the fanfiction, Naruto was told false stories about his mother that made him dislike her. For example, if he was told she abandoned him for some untrue reason.