ball
鞠 centers on a ball, especially one made of leather. The meaning is concrete and specific, referring to the object itself.
鞠 combines 革 (leather) with 匊, which likely contributes the sound. The character originally referred to a leather ball used in ancient games.
The left side 革 means leather, and the right side 匊 looks like a ball wrapped in a hand. Picture a leather ball being cupped in a hand.
For キク, imagine a leather ball that makes a 'kick' sound when struck: kick -> キク, and the ball is made of leather.