Contrary to what some may think, mutations are not a combination of the First and Second Mind, instead they are both completely independent when mutating.
The First and Second Mind Magic are both going to be able to mutate, while the Third Mind Magic cannot mutate at all. If the player wishes so, they may choose to not mutate their magic to learn additional moves for their source magics.
Mutations that are listed in the Official Arcane Adventures Wiki which are likely to be implemented in the future include: Phoenix, Sun, Plasma, Gravity, Shatter, Acid, Crystal, Metal, Equinox, Flash, Aether, Magma, Sand, Ash and Explosion.
Verdies, Angry Pirate Captain - Plasma Magic
Theos, The War Phoenix - Sun Magic and Phoenix Magic