The under part of my kitchen sink is flaking off some white particles
The white material that is flaking off looks like an insulation/sound deadening coat applied by the sink manufacturer. It wasn't supposed to come off. Unfortunately, I don't think there is a quick &amp; easy solution. 

The white material looks friable (it breaks up into dust). I see the white dust on your garbage disposal unit. If the sink is over a couple decades old, have a flake of the material looked at by a lab that deals with asbestos. Wet a paper towel, dab up a sample and drop both the sample and the paper towel into a zip-lock baggy. The lab will just look at the sample with a microscope, so it shouldn't be expensive. Often they will take a look at no charge.

If it contains asbestos have an asbestos abatement contractor remove the sink. Again, don't assume their service is super expensive - abatements are very common.

If the material doesn't contain asbestos, here is a suggestion: Remove everything that is stored under the sink to enable your access. Wear respiratory and eye protection - a good dusk mask and goggles. Scrape the material off the bottom of the sink. It looks like it may come off fairly easy. The tool that comes to mind is a paint scraper. A dust management tip is to wet the white material with a spray bottle before you scrap it off. I would clean up with wet paper towels rather than a broom or a vacuum, just for better dust management.

After the bulk of the insulation is removed you could coat the bottom of the sink with an encapsulant, or even another sound deadening product.