One possible fanfiction could be that Kakashi stumbles upon Naruto crying alone in the forest. He sits beside Naruto quietly, not saying a word at first. Then he gently puts his hand on Naruto's shoulder and tells him that it's okay to cry, that showing vulnerability doesn't make him weak. Naruto then pours out his heart about his loneliness and how much he misses his parents. Kakashi listens intently and shares some of his own experiences of loss, creating a deep bond between them in that moment.
Kakashi might immediately offer a comforting presence. He could put his hand on Naruto's shoulder and just be there silently, letting Naruto know he's not alone.
Their relationship is like a mentor - mentee bond. Kakashi has trained Naruto, and when he sees Naruto achieve great things, it can be overwhelming. For example, when Naruto masters a new jutsu that Kakashi thought was difficult for him, Kakashi might cry out of joy for Naruto's accomplishment.
One possible reason could be that Naruto is feeling lonely. Despite having friends, there are times when his past as an orphan and the years of being shunned by the villagers resurfaces, making him feel isolated. Kakashi finding him in such a state might then lead to a heart - to - heart conversation about Naruto's inner struggles.
Kakashi would probably be very calm at first. He's an experienced ninja who has seen a lot. He'd gently pat Naruto on the back, offering silent comfort. Then he might start to speak softly, asking Naruto what's wrong in a gentle voice.
The plot might involve Naruto crying due to a misunderstanding with his friends. Kakashi witnesses this and steps in to mediate. He first calms Naruto down and then arranges for Naruto and his friends to have an open conversation. During this process, Kakashi shares some wisdom about friendship and communication, helping them resolve their differences.
Kakashi would probably first check if Naruto is hungry or needs a diaper change. If that's not the case, he might hold Naruto close and gently rock him while humming a soft tune.
Perhaps Naruto has just learned some dark secrets about his past, such as the truth about his parents' death. Discovering such painful truths could easily make him break down and seek comfort in Kakashi, who he looks up to as a mentor. Also, it could be that he's feeling lonely. Despite having friends, there are times when the weight of his responsibilities makes him feel isolated, and crying in Kakashi's arms is a way to let out those pent - up emotions.
Sakura could be the one to find him crying. She has known Naruto for a long time and they are comrades. If Naruto is in a vulnerable state, she might stumble upon him.
I'm not sure about a specific 'young Kakashi crying' fanfiction off the top of my head. But you can try looking on fanfiction websites like Archive of Our Own. There are a ton of Naruto - related fanfictions there, and you might be able to find one that has this scenario.