Can I put a cheesecloth over my water faucet to get rid of the possible bugs in NYC water?

A cloth on the faucet reduces the amount of copepods and can be used as a filter on a temporary basis. It is best to use several layers of cloth for extra protection.  Ensure that the cloth is tightly woven, otherwise some copepods can get through.
It’s important to clean the cloth regularly, especially when it gets brown, or even better — change the cloth completely, because the water pressure could push apart the fibers of the cloth, even if it’s a tight weave, allowing copepods to get through.