The mutate card has to be applied to a certain spell card, in this case a fire cat. If you know the fire cat spell, during battle, click the mutate card, then the fire cat card, and it will turn into a storm cat (or whatever the mutate card says). You can use it then or leave it in your deck for later.
If you DON'T know the fire cat spell, you won't be able to apply the mutate card yourself. Ask one of your friends who is fire school to do it for you and then give you the storm cat card back, or trade the mutate for a card you can use. Also, the mutate card can only be applied to a regular spell card, not a treasure card or a card from your gear.
One nice thing about mutated cards is they cost one pip less than the original. Fire cat costs one pip, but storm cat is zero pip. Fire elf is 2 pips, but Storm elf is only 1 pip, for similar damage.
Then you should be able to just combine the two cards. When it shows up in your deck from drawing it, then select it and then select the Fire Cat card. They should both merge together to form the spell. Then you can choose that card and attack an enemy, or leave it to make it a Treasure Card for later.
If it is not working that way, then perhaps it is a glitch. If you can't seem to find your mutate card, then maybe it is not in your equipted deck. Double-check to make sure.