Polish residue that has dried on the car too long is very hard to remove. Before you start, divide the car into five sections--hood, roof, trunk, right side, and left side--so you can apply the polish and remove the residue one section at a time without worrying about over-drying. Work on the hood, roof, and trunk surfaces first to minimize the chance of scratching the surface with grit from the side panels. (Even when the car has been properly cleaned, it's best to be on the safe side.)
To begin, fold a clean piece of terry cloth twice, or, if an applicator came with the product, use it. Folding the cloth will give you a nice, padded surface that you can turn to a clean side every time you start a new section.
Put enough polish on the cloth to get a nice, light, even coating (don't flood the cloth). Apply it to the surface by pressing gently while moving your hand in small, overlapping, circular movements. Add more polish to the cloth as needed, until the section you're working on is covered.