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.
It could be that Sakura betrays Naruto's trust. For example, if Naruto entrusts Sakura with a very important secret, like something related to his family or his new power he's developing, and Sakura reveals it to their enemies or uses it against Naruto in some way, that would be a huge blow to Naruto and could lead to him crying.
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 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.
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.
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 reason could be extreme emotional pain. Maybe he witnesses the death of a very close friend or loved one in the fanfiction world. This overwhelming grief could cause such a physical manifestation as crying blood.
It could also be related to personal relationships. Tsunade might have been hard on Naruto in a way that he felt was unjust. Perhaps she was overly critical of his unorthodox ninja techniques or his impulsive nature, and over time, this built up into hatred. Naruto, being someone who values respect and recognition, could have resented her for not seeing his potential in the way he wanted.
One possible reason could be to search for a powerful artifact. In many fanfictions, there are often hidden and powerful items that can change the fate of a village or even the whole ninja world. Naruto and Tsunade might leave Konoha to find such an item before their enemies do. This would be a dangerous mission, so they keep it a secret from the rest of the village to avoid panic.
One possible reason could be a misunderstanding. Maybe Naruto overheard some false information about Jiraiya and Tsunade that made him think they had wronged him or someone he cared about. For example, if he heard they were involved in a plot that led to the death of a fellow villager, it could turn his feelings sour.
Perhaps Naruto blames them for some of the hardships he faced. Maybe he thinks that Jiraiya wasn't around enough when he needed a mentor, and Tsunade didn't do enough to support him in certain situations. This could lead to feelings of resentment and hatred in the context of the fanfiction.