A possible plot could revolve around Kakashi's team members' reactions to Naruto being his son. For example, Obito's spirit (if it has any influence) might have mixed feelings. And in the story, Naruto might inherit some of Kakashi's unique abilities in a more natural way than in the original series. The fanfiction could also explore how the Akatsuki views Naruto differently, knowing he is Kakashi's son, and how that affects their plans to capture the nine - tailed beast within him.
Sure, there may be fanfictions with that concept. In the original Naruto universe, Naruto's heritage is clearly defined with his father being Minato. Kakashi plays an important role in Naruto's growth as a ninja, but not as a father figure. Fanfictions like this might be created to explore different emotional and character - dynamic aspects, but they are not based on the official lore.
Often, Gai is paired with Kakashi in Hurt Kakashi fanfiction. Gai, with his unwavering belief in friendship and his eternal rivalry with Kakashi, can be a great character to interact with a hurt Kakashi. He might try to pull Kakashi out of his slump through their friendly competition. Also, Rin is sometimes included. Since she has a significant past with Kakashi, her presence in the fanfiction can either be a source of comfort or more pain, depending on how the story is written.
One possible motive could be to help Naruto control the Nine - Tails' chakra better. Hypnosis might be a way to reach into Naruto's subconscious and find a way to coexist with the powerful force inside him.
Naruto would probably be one of the first to try and rescue Kakashi. He has a strong bond with Kakashi and would not stand by while his teacher is in danger.
Naruto is often a common character. Since Naruto has a strong bond with Kakashi, he might be involved in Kakashi's time - traveling adventures. Sasuke could also be there. Their relationship with Kakashi in the original story makes it likely for them to show up. And of course, Guy would be a great addition. His friendly rivalry with Kakashi would add some humor and action to the time - travel scenarios.
😋I recommend 'The Three Swords of the Hatake Clan' to you. The protagonist of this novel transmigrated to the Naruto World and became the eldest son of the Hatake Clan, Kakashi's older brother. He had blended into the Naruto World and fought to protect his family and friends, striving to bring eternal peace to the Ninja Realm. I hope you like this fairy's recommendation. Muah ~😗
Kakashi would likely be shown as a loving and protective husband. He would still be the skilled ninja we know, but he would also have a soft side for his OC spouse. He might share his past traumas with her and let her in on his more vulnerable side.
Another reason could be related to Kakashi's past actions. If there are some secrets or things Kakashi did in the past that come to light and Naruto doesn't like them, he could snap at Kakashi. Naruto has a strong sense of justice and loyalty, so if he feels betrayed or misled by Kakashi, his reaction could be to snap.
Kakashi would likely still maintain his cool and mysterious demeanor. He'd be the experienced ninja, quick to analyze situations and offer sage advice. For example, when faced with a difficult opponent, he'd be calm and strategize on the spot. Hinata, on the other hand, would be her shy yet determined self. She might blush a lot around Kakashi at first but would show great courage when it comes to protecting others. Her Byakugan would be a crucial asset in the story, like if they were searching for something hidden in a lemon grove, she could use it to detect any hidden threats.