When claying, you gotta consider how it works, the clay picks up the contamination and *then the clay is contaminated*. Some of the contamination is embedded in the clay, but some of it will protrude from the clay and act like sandpaper. If you keep moving that clay across the surface you'll cause scratches. So as soon as the clay picks up some potentially abrasive contamination, you gotta knead/fold/replace the clay. Sometimes this means you must do something after *one* stroke with the clay. Takes a lot of self-discipline and patience, but that's the only way to be truly safe about it. Many people might be better off with a middle-of-the-road approach, trying to minimize the marring but accepting that they'll probably have to polish out *some* scratches.
FWIW, I like the Sonus clay from the Autopia store but I can't tell you if it's inexpensive or not.