This side quest has you chasing down a powerful shade that blocks a bridge. This NieR Replicant Ver1.22 A Bridge In Peril Guide will tell you where the shade spawns so you can keep your eyes on the location as you continue to explore and progress through the game.
NieR Replicant is filled with side quests. Some of them are easily done and complete in just a few minutes. However, others take a bit more time and effort to complete. A Bridge In Peril is one of the quests you can find in your own village, one of the villagers asks you to kill a Shade that nearly killed him and his wife.
NieR Replicant Ver1.22 A Bridge In Peril Guide
The bridge you need to check is the one to the West. It’s between the entrance to The Aerie and your characters village. I came over this bridge many times but never saw the enemy, so it’s not a guaranteed spawn. However, I did complete The Aerie area of the story, and on my return it appeared. Again, I’m not certain that’s a trigger but you can at least know the exact location so you can just check it running through.
There is a lot of backtracking at this part of the story so I would avoid using the fast travel options and just keep going through the region until it spawns.