I have no clue I need to see the vanity and understand why the TP holder is there,

I would think that you could cut think plywood and liquid nails over the top and then paint it.

My first question would be why not go back to a tissue holder with two posts that cover the same spots? Alternatively you could get one with a back plate.

If you are set on this holder just paint the whole side corner to corner. Use a shellac primer and 220 grit paper to make it smooth. Paint with a semi gloss latex white and a fine nap roller.

Patricia,It is rather confusing why you would bother to answer if you have no clue?

CatBuilder, yes, I need a good primer, that was the problem. why the paint did not work well over those spots. Thanks!

Thin sheet of ply and repaint

If you add plywood to the side, it will likely extend past the face frame and toe kick, and result in an unsightly view of the edge of the plywood from the front.

How about a 3-4" wide strip of 1/8" wood, with the edge sanded really well, and painted to match the vanity. Remove the TP holder, glue the strip on over the top, and re-install the TP holder. Then it's a "decorative strip" instead of a tacked-on side wall.

You mention the side is "not wood that takes paint well" so what is the side? I would have guessed MDF with factory paint. Usually you can find a primer designed for the material, give it a good scuff sand to knock down any shine and prime and paint. The people at the paint departments or stores are often helpful in selecting the right primer and paint for the job.

You could create simple moulding and frame out the side of the vanity.

Maybe painted bead board?