I would imagine that in-universe, some of her friends call her Karen, and some call her Kara, depending on when they got to know her and the situation. And that one of the two Supergirls is called "Car-uh" and the other is called "Care-uh". Supergirl for example has been called bother versions of the name in animation and live action. What's Supergirl going by as her civilian name these days?