The Pegasus band of faucets uses a single handle to control either one side (hot or cold water) or both sides of your water system. The handle can break or sustain damage because of corrosion and hard water stains. If this happens, the best answer is not cleaning it but replacing it. Find a replacement handle that matches your faucet model and simply install the new handle. It requires no plumbing skill at all, just a basic knowledge of the simplest of tools and about 15 minutes of your time.
Turn off your faucet water supply by closing the home's main water lines or shutting both valves under the sink.
Shut the drain by pulling up the stop or cover the drain with something to prevent parts from slipping down into it during repair.
Locate the cover on the handle of the Pegasus faucet and take it off by prying it up with your fingers. Set the part aside. If the part does not com up easily, use the edge of a knife to pry it up.
Unscrew the retaining screw, located on the top of the handle right in the center, with a Phillips screwdriver. Set aside the screw and pull the handle off the base of the faucet by hand.