1. To clean (some surface) vigorously or continuously with a sponge. I sat there sponging away at the concrete until all the markings were removed.
2. To remove something by absorbing or wiping it with a sponge. In this usage, a noun or pronoun can be used between "sponge" and "away." Dad's out back sponging the graffiti away. The nurse sponged away the blood from the woman's shoulder.
sponge something away
to absorb, wipe up, and wipe away something, as with a sponge. Try sponging the stain away with some soda water. I will sponge away the mess.