Well, perhaps Hinata is jealous. She might see Sakura getting trained by Kurenai as an advantage that Sakura will gain over her. In the world of fanfic, there could be unspoken rivalries or insecurities. Maybe Hinata had hoped to be trained by Kurenai herself and feels left out now.
In fanfic, there could be a backstory. Maybe Hinata has some history with Kurenai that makes her sensitive to Kurenai's actions towards others. And when Kurenai starts training Sakura, it triggers those negative feelings in Hinata. It could be that in the past, Kurenai made a promise to Hinata or Hinata's family about something related to training, and now by training Sakura, Kurenai seems to be breaking that unspoken agreement.
One consequence could be a strain in their relationship. Hinata might start avoiding Kurenai, which could create an awkward atmosphere. For example, if they are in the same team or group, it could affect their teamwork.
Obviously, the main characters are Kurenai and Sakura. Kurenai is the trainer and Sakura is the one being trained.
The fanfic could start with Sakura realizing she has weaknesses in certain areas of her ninja abilities. Kurenai notices Sakura's potential and offers to train her. During the training, Kurenai not only focuses on physical skills but also on mental strength. They might encounter some enemies who underestimate them at first but are then defeated by their combined efforts. This could lead to Sakura gaining more confidence and respect within the ninja community, all thanks to Kurenai's training.
Well, it could be centered around the training of someone who has a special connection to sakura. For example, a ninja in a fictional world who uses sakura petals in their combat techniques and is undergoing intense training to master new skills. The fanfic would then detail their training journey, the challenges they face, and how the sakura elements play into their progress.
A hinata and sakura lemon fanfic might explore the relationship between Hinata and Sakura in a unique way. It could be about their friendship growing in a lemon - scented orchard perhaps. Maybe they discover a new type of lemon together and that experience bonds them more deeply.
One possible reason could be for the sake of creating a more complex love triangle in fanfiction. Maybe the writer wants to explore a different dynamic where Naruto and Hinata's relationship is strengthened by having a common dislike for Sakura. It could also be that the fanfiction writer is trying to give Hinata more agency and make her a stronger character by having her oppose Sakura in some way, perhaps in matters related to Naruto's affections.
Some people might write them out of jealousy. For example, if they prefer other characters and feel that Naruto, Hinata, or Sakura get too much attention in the original series, they might write bashing fanfics. It's a rather immature way to deal with their feelings though.
Maybe in the fanfic, Kurenai has some actions that conflict with Naruto's beliefs. For example, she might oppose Naruto's way of handling certain village matters.
Some people might write such fanfiction because they have a different view of the characters. They may not like Hinata or Kurenai for some personal reasons and use fanfiction as a way to express their negative feelings.
It really depends on the specific fanfic. It could be a young girl named Sakura who is training to become a powerful mage. She might be the main character, with her friends and mentors being secondary characters that support her in her training journey.