Maybe he's trying to hide his identity for a secret mission. In some fanfictions, Naruto might be undercover, and covering his face is a simple way to avoid being recognized.
The Joker cutting his face off is a symbolic act of completely letting go of any pretense of normalcy and embracing his insanity to the fullest. It's a shock tactic to terrorize and confuse his enemies.
Perhaps he's trying to protect himself from some kind of danger. There could be an enemy who is looking for him and by covering his face, he can avoid being detected easily. It could also be a part of a new jutsu or power he has discovered where covering his face is necessary for its activation or control.
Well, it could be that Kakashi, being a character who values the integrity of the original Naruto narrative, sees fanfiction as a deviation. Fanfiction might include out - of - character scenarios for him or the other characters. He might also dislike the fact that some fanfictions can be of very poor quality in terms of writing and plot, which could be a reason for his dislike.
Maybe he thinks fanfic often distorts the real story or characters of Naruto. In the original Naruto series, Kakashi has his own understanding and relationship with Naruto. Fanfic might create some scenarios that go against his view of Naruto or the world they live in.
It could be due to physical exertion. If he has been running or doing some intense exercise, blood rushes to the face, causing it to turn red. This is a normal physiological reaction as the body tries to cool down.
His face might get red because he is embarrassed. For example, if he makes a mistake in public or is caught in an awkward situation, blushing can be a natural reaction.
In this fanfiction, Kakashi could help Naruto in his training through mental conditioning as well. Naruto is often impulsive, so Kakashi may work on his patience and decision - making skills. He could use his Sharingan (if applicable in the fanfiction) to analyze Naruto's movements during training and give him detailed feedback. Moreover, Kakashi could introduce Naruto to different types of enemies and battle scenarios in a controlled training environment, which would broaden Naruto's understanding of combat and make him a more versatile ninja.