I wonder what is the reason that causes the black patches in the result using cycles renderer with smooth shading? Is there any other way to fix them without making more subdivisions (The black blotches disappear when I use 2x subdivision surface)?
There is an option to get rid of those shadows by smoothing the mesh. Reason of those shadows is hard angle between faces.
Main downside of it is loosing some details along the mesh, but you can try and smooth only needed "patches".
In my example I've smoothed whole mesh by Smooth Vertex 5 times.
You can also Subdivide Smooth selected faces, but this will generate tris/ngons around selection.