A student of mine is doing a project on this topic, and I have realized that I cannot answer the question.
If we are talking about dropping the ball, and assuming a completely elastic ball, it should bounce back to the original height. Therefore, buy adding pressure, you are increasing the elasticity of the ball. But why does pressure affect elasticity? Is it through the air inside the ball or through the ball material? If it's because of the material, why does it become more elastic when tension is increased?
When kicking a ball, if it's always kicked with the same force, the change of momentum of the ball will be higher if the impact time is longer. If I think about it as an adiabatic gas under an extra pressure $\Delta P$, the new volume would be
$V =V_0 \left(\frac{P_0}{P_0+\Delta P} \right)^{1/\gamma}=V_0 \left(\frac{1}{1+\frac{\Delta P}{P_0}} \right)^{1/\gamma}$
For a small $\Delta P$, $\frac{\Delta P}{P_0} << 1$, I can do a Taylor expansion and I would have:
$V = V_0 \left( 1-\frac{1}{\gamma} \frac{\Delta P}{P_0}\right)$
I will assume that the kicking force (and therefore $\Delta P$) is the same in all cases. Therefore, the higher the initial pressure, the smaller the change in volume, and the impact time is shorter and the change in momentum should be smaller. Does this make sense?
Everything here seems really obvious and I have the feeling that the answer is really stupid. But I cannot find a good way of explaining it. A reference that I can give my student would be much appreciated.