He could also be hiding his injury to avoid being a burden. In the world of the fanfiction, his comrades might already have a lot on their plate, and he doesn't want to slow them down or cause them extra worry with his injury.
He might use his clothing to conceal it. For example, if he has a cut on his arm, he could wear long sleeves or a cloak to cover it up.
Maybe Zuko has his own goals that don't involve chasing Aang. He could be more focused on redeeming himself in other ways, like helping the Fire Nation people or learning about different cultures without the need to pursue Aang.
Maybe he wants to blend in better. If he's in a situation where he needs to gather information undercover, showing off his skills would blow his cover. So he hides them to seem like an ordinary person until the right moment to strike.
There's a possibility that Zuko would meet his grandfather. His grandfather's actions and decisions might have influenced the path the Fire Nation took towards war. By meeting him, Zuko could try to sway his grandfather's views and change the future of the Fire Nation at its root.
In fanfiction, she could be injured during a battle against dark wizards. Maybe she was caught off - guard while trying to protect her friends.
One possible reason could be that he is on a secret mission and doesn't want to be recognized. He might be infiltrating an enemy organization or spying on someone. Another reason could be that he has some new power or mark on his face that he wants to keep hidden until he fully understands or masters it.
Zuko's complexity makes him so interesting in these fanfictions. He has a troubled past which has shaped him, but he is constantly evolving. His ability to show vulnerability in front of the reader is another great trait. It allows for a deeper emotional connection between him and the reader, as seen in stories where he opens up about his insecurities and fears.
Maybe he wants to protect his loved ones. If his enemies know his true strength, they might target his friends and family to get to him. So, by hiding his skills, he can keep them out of harm's way while still being able to fight when necessary.
It could also be due to some sort of accident in the lair. Maybe there was an experiment gone wrong by Donnie. Mikey is curious by nature and might have been poking around where he shouldn't have, getting caught in an explosion or a malfunctioning device.
At first, Zuko might be hesitant to accept Hakoda's comfort. But as Hakoda continues to show kindness and understanding, Zuko could start to open up. They might develop a friendship based on mutual respect.