Quitting and leaving a mission in Nioh 2 is not what one would call a straight-forward affair, requiring an item rather than menu shortcut to perform. There are two items that will allow you to quit and leave a mission in Nioh 2, and one of them is always accessible.
Use either the permanent Divine Branch Fragment or single-use Divine Branchs to quit and leave missions.
Whenever you tap the touchpad on the DualShock 4 controller you will bring up the main menu, and the second icon (Items) is where you will find your way out of missions. Under Key Items is the Divine Branch Fragment and using it will take you out of a mission and back to the map. There is a cost to using the Divine Branch Fragment, however: all of your Amrita (experience in Nioh 2). This includes Amrita you brought into the mission with you, so it can prove costly.
If you’d rather quit without losing your experience you can use one of the rare Divine Branches scattered about the game. Again, these are rare, so you can’t stockpile these like you can other items. These are best used when you find yourself unable to finish a mission but have a veritable wealth of Amrita you can’t use to level.
All said, the game does save whenever you use a shrine or die, and there is always Rest Mode on the PlayStation 4, so I recommend only bailing out when you are in desperate need of a respec, or are unable to finish a mission despite your best efforts. Even summoning friends can be done within a mission, so popping either item should be used as a last ditch effort to return to the map.
That’s how to quit and leave missions in Nioh 2. It’s not difficult once you know what to do, but it comes with a heavy penalty. Abandon with care.