remove stains from paper photoshop cc 2021?

The area in the red circle doesn't really look like a stain, but more like smudged ink or something like that. You will most likely have to carefully clean that up with the paint brush tool and maybe create a selection with either the pen tool or the polygon lasso tool to help keep the lines clean when you paint the area.

The second photo shows using a second black and white adjustment layer. If you decide that the first b&w adjustment layer makes some parts of the image too light, because they are stained, but without the stain, they should be black, you can paint on the mask of the first b&w adjustment layer to bring back the darker luminosity. How6,doing that will also bring back the stain color. So the second b&w adjustment layer just keeps the image black and white and doesn't change the luminosity of the stained areas.

Here is the image with a b&w adjustment layer applied with the red and yellow boosted up. It does lighten some of the areas that actually should be black, so what you could do is add another B&W adjustment layer, and leave the settings alone, then paint in the areas that should be dark with a white brush on the first adjustment layer's mask.

Since it looks like the image is b&w, add a b&w adjustment layer then boost up the reds and yellows in the adjustment layer to lighten then staining.