Luckily, mesh fireplace screens are easy to clean.
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Wet a cloth and use it to wipe down the front and back of the mesh screen. Do this until you have removed most of the soot.
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Rinse your cloth out and then add a squirt of dish soap. Wipe the mesh screen down again with the cloth. Make sure you get both the front of the screen and the back.
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Rinse the mesh screen out by holding it under the faucet. This will wash away the soap and any remaining dirt.
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Dust the mesh screen in between cleanings so that the soot does not build up as quickly.