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To absorb or mop up some liquid with or as with a sponge. A noun or pronoun is used between "sponge" and "up." One of the worst jobs in the hospital is sponging up all the blood and other bodily gunk that ends up on the floors after surgeries and deliveries. You need to sponge that paint up straight away—if it gets dry, you'll never get it off the floor.
sponge something up
to absorb or take up moisture, as with a sponge. I had to sponge the spilled milk up from the floor, the chair, the table, and the baby. What a mess! Liz sponged up the water.