Yes, there are quite a few. You can check out some fanfiction platforms like FanFiction.net or Archive of Our Own. Just search for 'Ash keeps Togepi' and you'll likely find a variety of stories with different takes on their relationship and adventures.
Ash could become more patient. Since Togepi is a relatively weak and cute Pokémon at first, Ash has to be patient in training it. This would be a change from his sometimes impulsive nature in battles.
Well, the plot might be that Ash stumbles upon a lost Togepi. Since Ash has a kind heart, he takes it under his wing. They then embark on a journey where Togepi starts to show some extraordinary powers that no one has ever seen before. This attracts the attention of Team Rocket who try to capture it. Ash and his friends have to come up with clever plans to outwit Team Rocket every time they make an attempt.
In a different plot, Ash and his friends come across an injured Togepi. Ash nurses it back to health. Togepi then becomes attached to Ash. However, there is a prophecy that Togepi will one day have to make a great sacrifice to save the Pokémon world. Ash is determined to change this fate and goes on a quest to find a way to break the prophecy while having many adventures along the way.
First, understand the characters well. Ash is brave and determined, and Togepi is cute and a bit mischievous. Then, come up with an engaging plot, like I mentioned before. Also, use vivid descriptions of the Pokémon world, the battles, and the emotions of the characters.
One idea could be Ash and Togepi getting lost in a mysterious forest. They encounter strange new Pokémon and have to work together to find their way out. Another plot could be Togepi getting sick and Ash having to search for a rare herb to cure it. He meets different trainers along the way who either help or hinder his quest.
FanFiction.net is a great place to start. They have a large collection of Pokémon fanfictions, and you can easily search for 'Ash keeps Togepi' within their database.
There's a fanfic where Ash discovers Togepi has a unique power. It shows how Ash helps Togepi to master this power. They train together, and their bond grows stronger as they overcome difficulties like dealing with a group of mean - spirited Pokémon trainers who try to steal Togepi for its power.
Ash could be a Pokémon trainer with a different starting point in this fanfiction. Born in 1987, by 1997 he might have already made a name for himself in local Pokémon battles. He could have a team of Pokémon that he rescued from difficult situations, like a Charmander that was abandoned by its trainer. His story could revolve around giving these Pokémon a second chance and proving that Pokémon don't need to be 'perfect' to be strong.
One popular aspect of Alola fanfiction with Ash could be his growth as a trainer. In the Alola region, Ash had unique experiences. Fanfiction might explore how he further developed his skills, formed new bonds with Pokemon, and interacted with the local culture. For example, his relationship with Pikachu might be delved into deeper, showing how they overcame new challenges together in the Alola setting.