How do you clear the saved-games, high-scores, etc. from (some GameBoy) cartridges?

There is no universal way to do this with all cartridges.

Some games have some hidden command to erase data. An example is Pokemon Ruby (GBA) where you need to hold Select + Up + B to get an option to clear game data. In Mario Kart Super Circuit (GBA) you need to hold L + R + B + start when turning on the GBA.

If the game doesn't have the option to erase data (hidden or not), you have two dirty options:

  1. You can damage the save data, removing the cartridge or turning off the system when the game is trying to save data. Risky? Yes, you can damage your cartridge, DO IT AT YOUR OWN RISK.

  2. You can remove the battery inside the cartridge, this will lose all saved data. As you bought used cartridges, you can replace the batteries with new ones. This video shows how to replace the battery:

Worth it? It's up to you decide. Good luck.