Mysteries of Kyoto is one of the three story types in Onimusha: Way of the Sword, alongside the main Way of the Sword story and Chance Encounters. As Musashi advances the main story, he receives requests from a certain individual, and Mysteries of Kyoto are the responses to those requests.
Optional, Not Mandatory
Completing every Mystery of Kyoto is not required to finish the game. Only a few of them affect the main narrative, but completing any of them earns various rewards, which makes pursuing them worthwhile even when they are not strictly necessary.
Finding and Starting a Mystery
People holding clues relevant to a Mystery of Kyoto are marked with an icon above their head out in Edo-Era Kyoto. Speaking to one of these people is how their tale, and the associated request, is heard for the first time. The map also marks Mysteries of Kyoto clue locations directly, alongside markers for the main story and for Chance Encounters.
Achievements
Steam's achievement list recognizes both starting into this system, with an achievement for completing the first Mystery of Kyoto, and mastering it, with a separate achievement for completing all of them.
Tracking Progress
The map marks Mysteries of Kyoto and their associated clues with their own dedicated icons, separate from the icons used for the main Way of the Sword story and for Chance Encounters. The Progress screen, opened from the Main Menu, lists the completion status of every story a player has picked up, including any Mysteries of Kyoto still open, and lets a player set one of them as the current destination.
Practical Notes
Because all three story types can be followed at once, there is no need to set Mysteries of Kyoto aside to focus purely on Way of the Sword; checking the Progress screen periodically is the easiest way to keep track of which requests are still open, and speaking to any character marked with a clue icon costs nothing beyond a short detour.