An OC with a strong and independent personality would be great. For example, an OC who is a skilled ninja in her own right, like a master of stealth or a powerful genjutsu user. This way, she can match Genma's abilities and they can have an equal partnership.
Another aspect of the training could be mental discipline. Hayate may introduce Naruto to meditation - like techniques to help him focus his chakra more effectively. He'd also likely teach Naruto how to keep a calm head in the heat of battle, sharing his own experiences to illustrate the importance of mental fortitude.
Well, in many fanfictions, Naruto might use his chakra to heal Hayate. Since Naruto has a large amount of chakra and has learned some healing techniques over time, he could transfer his chakra into Hayate's body to repair the damaged parts.
One common theme is loyalty. Hayate is always loyal to Nagi in the series, and this is often explored further in fanfiction. For example, how far he will go to protect her.
Naruto might initially be a bit jealous that Genma is spending so much time teaching Sakura. But as he sees Sakura improve, he starts to respect the process. Genma, on the other hand, might see Naruto's potential through his connection with Sakura and start to offer some indirect advice to Naruto as well. Sakura would grow to have a new level of respect for Genma as her teacher.
One great 'Nagi x Hayate' fanfiction is 'A Chance Encounter'. It beautifully portrays the relationship between Nagi and Hayate, with vivid descriptions of their interactions. The author does an excellent job of showing Nagi's dependence on Hayate in a really heartwarming way.
Another problem might be the lack of proper resources. Even if Naruto has the ability to heal, he may not have all the necessary herbs or medical equipment. In the world of Naruto, some healing requires specific items. If they are in a remote location or in a battle - torn area, getting these resources could be a huge hurdle.
Well, 'sakura trained by genma fanfiction' might involve a story where Sakura is put through some form of training. Genma could be a person, a concept, or something else within the fanfiction world. Maybe it's a training in a special skill, like magic or combat, and the story could explore Sakura's growth and development during this training process.