I have and invented design improvement on a product that already has a patent on the product. Can I obtain my own patent for the new design on that product?
You must introduce some novel aspect to that product' design. You would file for a patent on that novel aspect. If the novel aspect has not been discussed before you have a chance of receiving a patent protecting only that novelty. You would be able to control how that novel aspect is used.
Whether you can market your product at all depends on the other patent holder. You would likely need to license that original patent. However, you should scrutinize the claims. The claims are protected, not the product. Your product may not infringe at all, depending on their claims and your design.