In this particular case, it is fairly easy to make things better. Since the colors are fairly far apart from each other it is easy to isolate them and make changes to even them out. If this were a photo (for example) this method would not work.
- apply a "smart blur" to the image to smooth out the color fields
- choose "selective color" (under image > adjust) and then add yellow to "yellows (drop down item", blue and yellow to "greens (drop down item)", and magenta to "magenta drop down item"
- adjust the saturation by lower saturation slightly and raising lightness slightly to bring the colors back in line with the original scan.
- make a new blank layer on top
- use "select > color range", pick the black, slide the "fuzziness" to max
- with the selection enabled, flood fill on the new layer with black, deselect
- make new "select > color range" selection using black and about 60 or so fuzziness
- with selection enabled, apply a layer mask to the top layer (this knocks out some of the greys)
This takes longer to type than to do.
(NOTE: In the future, you might look into a photo stand and use a digital camera, photo lights, and a polarizing filter to cut glare.)