In some fanfictions, it could be due to a special jutsu gone wrong. Maybe a jutsu that Kakashi was using to transfer some of his chakra to Naruto for a mission accidentally transferred his Sharingan power as well.
Sakura could also be present. She might be observing the training and learning some things herself, or she could be helping Kakashi in some way, like providing medical support during intense training exercises.
Well, Kakashi might use his Sharingan to intimidate the civilians and create an opening to grab Naruto. His quick jutsu skills could also be a factor, like using a clone to distract the civilians while he gets Naruto to safety.
He might learn some advanced chakra control techniques. For example, better management of his Nine - Tails chakra. Kakashi, being a skilled ninja, could teach Naruto how to blend his own chakra with the Nine - Tails' more effectively, which would enhance his combat abilities.
Kakashi would probably focus more on stealth training. In the original, Naruto was more about big, flashy attacks. But Kakashi would teach Naruto how to move quietly and strike unexpectedly. He might also teach Naruto more about chakra control from the very beginning. In the original, Naruto struggled with this for a while. And Kakashi would surely introduce Naruto to some of his own unique jutsu earlier, like the Chidori, but with Naruto's own spin on it.
Naruto might also try to find alternative ways to gain strength. He could explore the forest around the village to find new training areas or discover new chakra - related phenomena. He has a curious nature, and this could lead him to stumble upon something that gives him an edge in his training, like a hidden hot spring that has special chakra - enhancing properties.
Female Kakashi could have different teaching methods from male Kakashi. She might focus more on teamwork and communication skills in a way that is more in tune with Naruto's natural inclination towards making friends. Their relationship could develop with more of an emphasis on mutual understanding of each other's strengths and weaknesses. Naruto may also feel more comfortable sharing his feelings about certain female characters or relationships with her, which would be a departure from his interactions with male Kakashi.
Kakashi's parenting style would likely be more laid - back compared to some of the stricter ninja parents. He might give Naruto more freedom to explore his own way of being a ninja. For example, instead of forcing a particular fighting style, he would let Naruto experiment and find what suits him best.
The moment when Kakashi actually reaches Naruto and stops him from harming himself is crucial. He could use his speed and strength to grab Naruto's hands if he was about to use a weapon on himself. Kakashi's presence alone at that moment can be a shock to Naruto, making him pause and re - consider.