Yo! I'm just sharing my tips if you like to mine in cavern mines.
Sometimes, you really like to mine in cavern mines, then you spend a huge amount of time going through mountain mines just to search for one of them. After you mine the stuff out of them, you proceed to spend more time searching for more cavern mines.
Well, here's my tip if you're a cavern miner. When you find a cavern mine, just mine whatever you want to mine in them. Then, DO NOT exit the mine. Just exit the entire game, then log in again in a few minutes time. The cavern mine should reset itself, so you can mine again. These processes are repeatable!
Happy mining, caverners!