Yes, there might be. You can try searching on fanfic websites like FanFiction.net or Archive of Our Own. There are a lot of fan-made stories out there for various pairings, so there's a good chance you'll find one for Naruto and Mizore.
One common theme could be the exploration of their different personalities coming together. Naruto is very outgoing and determined, while Mizore is often more reserved. Their relationship could be about how they balance each other out.
As I said before, FanFiction.net and Archive of Our Own are great places to start. Just go to their websites, use the search bar, and type in 'Naruto x Mizore'. Then browse through the results to find a story that interests you.
A fanfic might explore the idea of Naruto being sent to the world of Mizore and Shirayuki for a mission. He has to adapt to their world while also helping them with a local problem, like a dark force threatening their land. This could involve a lot of character development as Naruto learns about their cultures and they learn from his ninja ways.
Another option is to check out Archive of Our Own (AO3). It has a vast collection of fanfictions. Use the search function and type in 'mizore x naruto'. You may need to sift through some results, but there could be some real gems in there.
One possible plot could be that Mizore, who is from a different world or dimension, somehow ends up in the Naruto universe. Naruto, being the curious and friendly ninja he is, decides to help her find her way back. Along the way, they face various challenges together and learn about each other's powers and cultures.
It could develop through a series of misunderstandings at first. Maybe Mizore misinterprets Naruto's friendly gestures as something more, or Naruto doesn't understand Mizore's shyness. But as they clear up these misunderstandings, they start to really understand each other. They find that they complement each other well, with Naruto's boldness and Mizore's calmness, and this realization leads to the growth of their relationship into a loving one.
Naruto would probably win in the long run. He has a lot of experience fighting powerful opponents. He's faced Akatsuki members and other strong ninja. His ability to adapt and learn new techniques quickly would give him an edge. Mizore is strong with her ice powers, but Naruto could potentially find a way around them. For example, he could use his Fire - Style techniques to counter her ice. Also, Naruto has the power of friendship and teamwork on his side which could give him an extra boost in a fight.
The fanfic could be about a dimension - hopping adventure. Naruto accidentally ends up in Mizore's world. Their first encounter is hostile as they don't understand each other. Mizore attacks Naruto with her ice powers, thinking he's an intruder. Naruto retaliates with his ninja techniques. But as they interact more, they realize they have more in common than they thought. They could be on a quest to find a way to send Naruto back home while also dealing with threats in Mizore's world.
You can try fanfic websites like Fanfiction.net or Archive of Our Own. They have a large collection of fanfics, and you can search specifically for Naruto and Mizore ones. Just use the search bar and type in the relevant keywords.