If you hide a part, you may still see pieces of it when using a cutting plane. Luckily, the fix for this is pretty simple.
There is a property on CuttingPlane to control if an invisible part should appear in a cutting plane:
CuttingPlane.computeFromVisibleParts If you change the property to hide the part, then it should stay invisible when the cutting plane is used.
The same applies for isosurfaces and isovolumes.