Eliminate speckles from scanned sheet music

In Photoshop, go to Image => Mode => Monotone

Go again to Dropdown list - Image => Adjustments => Curves.
A pop-up box will show up with a slanted line going to the top right hand corner.
Pull the edges horizontally towards the mid-point on their respect x-axis. In the "Input" box, the top should read 50 and the bottom should read 49 (as close as one gets).

For larger areas of speckles, use the eraser tool to erase them or Brush tool to paint it all white.

For smaller speckles that's tedious to clean, use the following filter.
Filter => Noise => Dust & Scratches.
I set my Radius between 2-4 and Threshold to 0.

I got a few more tricks up my sleeve. but here are the basics. Hope it helps!

Maybe you could use a sheet music OCR program, and "regenerate" it.

It looks like you have made some adjustment to your scan that has intensified  your problem or you use scanner settings that increased the contrast of the dirt like some threshold adjustment.

The difficulty is removing the unwanted marks whilst keeping the small wanted marks such as staccato dots or slurs. To Photoshop both are just white or black pixels.

The best I could achieve was duplicating the image, blurring it slightly and using Linear Dodge blend mode. Then adding a curve to restore some contrast.

Dave