You can upgrade a charm by leveling it up to level 30. All you need are Charm Points and some coins.
You can also merge two identical charms to improve one of them:
If both are the same charm → the main one gains +1 potential power and gets a stat boost.
If the charm you’re using as material already has potential → adds +2 (or more) improvement power, depending on how upgraded it was.
The 3 sockets to the left of the Puzzle are the charm’s improvement potential.