I wonder which type of kicking may cause the ball to go a larger distance? One way is kicking the ball when it is at rest and another is kicking a moving ball in the opposite direction. If the ball goes farther in the second way,then what is the reason despite the fact that a moving ball will exert extra force to the player's leg?
When you look at this video you can see that the lower leg seems to maintain a constant velocity. This is probably partially due to the higher total mass of the leg compared to the ball. The ball leaves the foot at a higher velocity. This due to the deformation of the foot and the ball. This deformation is caused by acceleration (initial velocity difference). When the ball tries to return to its original spherical shape, it will push against the foot, which pushes back and accelerates the ball.
Now if the initial velocity difference would be bigger, due to the ball moving towards you, the deformation will be bigger. Assuming that the leg still can maintain roughly a constant velocity, this means that the ball will be accelerated for longer and thus have a higher velocity.
And other way to look at this is that the ball bounces of a wall. However in this case the wall is moving. Relative to the wall the ball reverses its velocity normal to the wall, but with a slightly lower amplitude due to inelastic collision. But this means that if the ball initially is moving faster towards the wall the ball will move away from the wall faster. This can be done by moving the wall faster (kicking with a higher speed) or by increasing the incoming velocity of the ball.