You can apply a polish without using the clay bar after a wash. Its more safe to use the clay bar inbetween because you'll remove more contamination with the clay bar. This means there is less of a chance of a large piece of contamination becoming lodged in your applicator pad, which could be grinded in to the clear coat. To remove more of those swirls and not just cover them up you might need a more aggressive polish. Let me know if you'd more information on polishes that will help remove more imperfections and help restore that deep shine.
You can use those yellow foam applicators to apply the polish but be careful with the edges. Sometimes those pads have a thick and firm edge or seam to them which could add fine scratches to the clear coat. As for removing the product you can use those MF towels. Just check the edges and seams for any rough material. Some MF towels use nylon threading or other improper materials that may add scratches. I also rip off tags if there are any on the towels. To test the towels you can use the back of an old CD and wipe the towel across the CD. If it leaves fine scratches its not safe and if not then it should be safe on the clear coat.
Good luck and let us know if you have any other questions.
Greg @ Detailed Image