How to separate two points in a drawing?

Whenever I cleanup the contours of a drawing, I sooner or later end up with points that seem to be on the exact same spot, and I cannot seem to find a way to separate them. Sometimes I can solve the problem by overpainting part of the contour, and flattening/optimizing the drawing, but every now and then I get stuck on a contour where this doesn’t work, making cleanup impossible. How on earth am I supposed to separate these points, when I can only select the two, or none?