Alright. Here's how I did it. Steps are the way to go, YouTube is too laggy for my connection.
- It's not in the cave, firstly. After the badge check, you will find a makeshift Pokemon Centre right of the cave entrance. Stand in front of it, preferably the first spot in front of the doorway.
- Go right. Keep going. You will find a small, hidden passageway. Follow it, and you will find a small area with flowers and an item. That item is the TM01!