A Request from Mai Mission Walkthrough - Pokémon Legends: Arceus
You should now go and complete research tasks to increase your rank.
The next morning, you'll talk to Rei/Akari and get a new mission.
Go to the Training Grounds and defeat Rei/Akari.
After the fight, you will learn about Type Matchups and Agile and Strong Style moves.
Go to the Obsidian Fieldlands. There you'll find Voto again.
He will teach you how to capture Pokemon by moving behind them and hitting them. This is an advantage in battle because the other trainer will assume you are attacking from the front, and may not be prepared for an attack from behind.
To prove your strength, go meet Mai and have a Pokemon battle with her.
Mai gives Vivichokes after the battle.
Go to Deertrack Heights. Talk to Mai.
Once you're healed, challenge the Alpha Kricketune.
Alpha Kricketune will be at level 12, but you should take your time to strengthen your team so that you can take it down. Once you reach the level cap, it's generally a good idea to further strengthen your team.
You can also try catching Alpha Kricketune, once you wear it down.