Pretty much what Aaron said. Clay will remove contaminants from on top of the surface, but won't do anything to fix swirls or scratches. I've used it to remove paint overspray and rail dust pretty successfully. You can also use an iron remover as a second step. Just spray it on, wait for it to turn purple and run off, then wash again. Just make sure you do that in the shade as you don't want it to dry.
For removing swirls, get a good polish and a DA buffer with the right pads. It can be intimidating and expensive to get setup, but the results are worth it.
As for protecting the pain after doing a correction, there have been a lot of new products on the market that are actually really good! From the $14 bottle of Meguiars hybrid ceramic (lasts about 6 months), to the slightly more expensive 303 graphene spray on coating (about 9 months), all the way up to true ceramic or graphene coating (typically 3-5 years, up to 7 with proper maintenance). Can be as easy as spray it on while the paint is wet and rinse again, or more like a traditional wax.
Use a pH balanced soap after applying any coating, as a non-pH balanced soap will strip it off. Some chemicals will also strip a coating.
I'm not sure what we drove through, but there are spots on the hood that do not have the same hydrophobic properties as the rest of the car. Best I can tell is that something hit the front of the hood and then ran up toward the windshield. Can't tell with the naked eye when dry, but you can certainly tell when it gets wet.