I'm not a physicist, but I'll take a stab at it.
A simplified example of your spinning sphere that may help you with this concept would be a disk made all of one density of material. An example would be a child's top or a gyroscope that you can spin on a flat surface. Every part of the disk has a matching balancing part on the opposite side of the disk. Each balancing pair of parts of the disk have the same mass as each other, have opposite motions to each other when rotating and create opposite balancing centripetal forces that keep the disc's rotation balanced around the center of mass (which is also the disk's geometric center).
If you add more mass to the disc anywhere but at the center, the center of mass of the disc shifts away from the geometric center of the disk and toward the mass you just added. The object will now rotate around this new center of mass. This is because all the mass on the side away from the new added mass must create a balancing opposite force to the now heavier side of the disk. The mass of the disc between the geometric center of the disc and the new (shifted) center of mass shifts to becomes the opposing balancing force to the added mass.
The picture below may help you visualize this:
The green dot on the right is the original center of mass and center of the disc. The blue circle is an added mass. The green dot on the left is the new center of mass. The area between the two red lines is mass on the disk that balances the added mass when rotating. Adding more (blue) mass will shift the center of mass further from original center and move the left red line (and center of mass) further toward the added mass (left). If the original disk was very massive relative to the added mass, the center of mass won't shift as far (i.e. less area between the red lines needed to balance the new mass, and less shifting of center of mass to balance the added mass).
So to conclude, every time you add to (or subtract from) the mass of a rotating object, the object changes the location of it's center of rotation so that the forces caused by rotation remain in balance. The point of rotation is the center of all the mass of that object.