It's possible that Adrian is trying to catch Rivky and Amu because they have something he needs. This could create a conflict as Rivky and Amu might not want to give it up easily. For example, if it's a magical object or a piece of information that is important to them. So the struggle between Adrian's pursuit and their resistance forms the main conflict.
Since I don't know the details of 'adrian catches rivky and amu fanfic', I can only speculate. Maybe Adrian is a character with a certain goal or mission, and Rivky and Amu are either his targets or his companions in an unexpected situation. It could be an adventure - filled story where Adrian has to 'catch' them for a good reason like saving them from danger or getting them to a specific place.
Well, it could be about Adrian catching Ricky and Amy in a compromising situation. Maybe they were sneaking around doing something they weren't supposed to, like planning a secret party or hiding something from Adrian. And Adrian's discovery sets off a chain of events, like an argument or a big misunderstanding that they all have to work through.
The main characters are clearly Adrian, Ricky, and Amy. Adrian seems to be in a position of discovery, while Ricky and Amy are the ones being caught by Adrian.
Maybe the fanfic starts with Adrian being a pro at breakdancing and Marinette being a beginner. Marinette is in awe of Adrian's skills. He offers to teach her. As they have their breakdance lessons, there are cute moments like when Marinette keeps tripping and Adrian catches her. Slowly, Marinette improves and they end up performing together at a local event, impressing everyone there.
Perhaps the main conflict could revolve around the prince's internal battle. His anxiety makes it hard for him to trust his logic when dealing with moral issues. Say, there's a situation where the prince has to decide whether to go to war for moral reasons (like protecting the innocent in another land) but his logic tells him the cost in lives and resources is too high. His anxiety then amplifies the struggle between what he feels is morally right and what seems logically sound.
Maybe in the fanfic, Ash hears about a powerful Mightyena causing trouble in a certain area. He goes to investigate and discovers that the Mightyena is actually being controlled by an evil force. Ash battles the controlling force and after freeing the Mightyena, it becomes his ally.
In the fanfic, Absol might be in some kind of trouble that Ash has to rescue it from. After that, Absol decides to join Ash's team. Ash would then have to train Absol and figure out how to best utilize its skills. They could go on adventures together, facing various challenges and meeting new Pokémon along the way. Absol could also introduce Ash to new areas or Pokémon that are related to its own origin or habitat.
The main conflict could be Merlin's struggle between his loyalty to Arthur and his duty to the magical community. He loves Arthur and wants to protect him and Camelot, but he also has a responsibility to the other magic - users who are often persecuted.
In the 'amu gets attacked by x eggs fanfiction', Amu herself could end up saving herself. She could find an inner strength or a new power within herself during the attack and use it to defeat the x eggs and get out of the dangerous situation.
A plotline could be that Amu and Tadase get transported to a parallel universe where their roles are reversed. Amu is now the cool and confident one while Tadase is the more shy and reserved. This would create an interesting dynamic as they have to adapt to these new personalities and still find a way to get back to their original world.