Usually you can use the "clone stamp tool" in Photoshop to do some touch-up; it copies some parts on the image and duplicates it where you want to. But for this you need to actually have something to copy.
In your image, it's not about removing dust; that's pretty much all there is in the holes. It's about digitally creating new holes and filling them with a new texture to make them look cleaner! There's many ways to do this such as selecting each hole, filling them with a new color, adding some shadow and gradients, etc. The more time you'll spend on it, the more realistic it will look, and it also depends on your Photoshop skills. I think it would be nicer and faster to fill the holes with new color and shadows than trying to remove the dust.
To make this image an illustration, you could import it in Illustrator and simply trace manually the shapes on top of your image. Once you're done you can get rid of the picture and adjust your vectors and gradients to look realistic.
You could always try the "trace" option; it will do this almost automatically for you but the results are not always very nice, especially since your image is quite symmetrical (eg. your circles might not look very soft and might have some corners!) So the best is probably to get to work, and use the basic shapes in Illustrator... and re-draw that illustration manually.
If you don't want a vector illustration but only want your image to look like an illustration, there's some filters you can explore in Photoshop that can help. You can also purchase add-ons or extra filters for this. Results may vary and you might need to try a few before you find something that works well for you. Search online, there's Adobe that offers some, but also other non-Adobe websites.