I agree with Karen. Your answer is "Yes." You can get a patent on new and non-obvious modifications of known products. And, as Karen noted, your patent might cover your modification beyond just a particular known product if circumstances permit.
Nobody in the US (or where ever your patent is) can use your modification without your permission.
The issue of blocking patents doesn't come up unless someone else has a patent on the product that you've modified. You wouldn't be able to make your own invention because of the patent on the underlying product. Thus, you both would be blocked by each other's patent.