A rescue piss happens when you take a dump and there is a piece of turd left on the porcelain that does not flush. Later on, you come back to take a piss and you send the turd to it's watery grave from the force of your piss, washing away the previous sins of leftover turb goblins.
I took a massive shit that left a piece of turd on the porcelain. I came back later and offered a rescue piss to force it into the water.