I fill a mug with apple cider vinegar and put it in the sink. Then I pull out the sprayer and stick it head-first into the mug and let it soak overnight. Usually, it's fairly clean the next morning and a little scrub with an old toothbrush gets the rest of the gunk off. I put apple cider vinegar in a plastic bag and stick it over my shower head and fasten it on with a rubber band. I let it soak as long as possible to loosen the build-up, then use that old toothbrush again. Good luck.
Use the same recycled and filtered white vinegar that you use for cleaning the coffee pot. (You DO clean the coffee pot, don't you?)
I use white vinegar for clogged faucets, sprayers, shower heads, soap scum on the fiberglas shower enclosure, and to clean the coffee pot. My woodwork is dusty, but my coffee pot is ALWAYS clean!!
Test a particle of the white residue in CLR or white vinegar. It could be calcium. If the crystals dissolve try soaking the spray head in white vinegar. I have seen particles of white plastic from the dip tube from a hot water tank that is disintegrating too. If the particles do not dissolve it could be the cause however these usually show up in the faucet aerator as well.
Sometimes I take a wire brush and push the bristles into the sprayer holes and then spray with hot water. I have even had to unplug the holes with a needle.