They're just CRAM dots. If I recall correctly, every time a new palette entry is swapped in, a dot (or multiple dots) will show up somewhere on the screen, so it's simply a matter of whether or not the developers have tried to move them to a less conspicuous portion of the screen. This will also happen with the original carts and has nothing to do with the Everdrive. The Sonic games are well-known for doing this when water is on the screen (they change palettes mid-screen to effectively double the number of colors that can appear at one time), leaving a waterline consisting of these garbage pixels.
The only way to get away from them is to play via emulation.