This is a perfect example of why not to use an acid-free paper. The circle on the right Batman's face is from using my spit to keep the paper stuck to sheet of glass while I coated it. I guess my spit is acidic enough to change the paper. The tones on this spot are wonderful.
Try adding a teaspoon or two of vinegar to the wash water when developing the cyanotype. That should offset the alkalinity of your tap water enough to get that "spit spot" tone over the whole print.