The air from the air pump doesn't actually go through the sponge... so the air pump will not "refresh" the air bubbles in the sponge...
I agree with Pharaoh, when the sponges are new/stiff they hold air bubbles... as they break in and soften up those bubbles escape and they stay down much better... Don't hesitate to keep on squeezing it to help soften it up...
Most suction cups don't work very well in aquariums. In my experience it's just a cheesy way of attaching something. The higher grade sponge filters have weights designed in them to keep them down...