Forgive my questioning here as it may have been stated already but are these spots raised, as in can you feel them OR are they level with the paint ?
The reason I ask is before you now how to remove them you need to know more about them. If raised this means they are an attachment of some sorts to your paint, as in a foreign material such as road paint, sap, glue, etc.
If there is no feel to them then it's most likely a stain of some sorts from a solvent, such as a few specks or a light splash of brake fluid or oil.
If you can feel the bumps but can't remove them with your fingernail then you may want to now try and dissolve them using something like turpentine, rubbing alcohol or Goo-Gone. Turpentine works great for sap as does alcohol.
I had to wet sand a black Mercedes once because nothing would remove some paint over-spray which come to find out was from a boat being painted nearby where they had parked for a boat cruise...tough stuff that boat paint!! So this may be something that needs to be wet sanded then polished back out.
For a solvent type stain there may be nothing you can do as the stain has "set" in the paint. Many solvent based tire dressing can do this as they sling off from your tires and come to rest on the lower plastic areas which are painted differently. This is most noticeable on light colored cars. Many times these pieces must be repainted (and that may not fully solve the problem) to solve that issue.
I highly doubt clay will help here nor would I seek to abrade these spots away. You first need to try and diagnose just what the stain or material is as this will help determine how to best remove it. Go to Radio Shack and for about $10 buy a lighted magnifying tool. You can place this on your paint and get a nice view of just what this material might be.
Anthony