That the underinflated ball goes further is a surprise to me (and my soccer-playing daughter). There are two follow-up questions that you might investigate:
i) Does it go further because the kicker can transfer more energy to it?
ii) Does the underinflated ball travel further than the fully inflated one, even if they both start with the same amount of energy?

You could answer the second question by launching the ball from a big slingshot -- perhaps something made from bicycle inner tubes fastened between two posts. Pull the slingshot back a fixed distance (perhaps marked by a peg in the ground) and make ten tests with each ball, recording how far they go. If the underinflated ball goes further, then this suggests the answer has something to do with the air resistance to each ball.

If the underinflated ball doesn't go further with the slingshot test, and yet it does go further when `kicked' by the kicking machine, then the answer must lie in the interaction between the boot and the ball. You might try making a series of tests, starting with a completely deflated ball, then pumping it up till it's almost bursting, and make a graph of the distance travelled versus the pressure in the ball.