Was American imperialism a continuation of Manifest Destiny?
I would say that it was, though there are differences.
Manifest Destiny was the idea that the US was destined to expand (at first, from "sea to shining sea") because of its superior culture, its superior religion, and its racial superiority. This was one of the major ideas that was involved in American imperialism as well. This is very similar to the idea of the "white man's burden" that helped to motivate American imperialism. The US felt that it deserved to take places like the Philippines and Hawaii because Americans were superior to the natives of those places. American imperialism may have had a somewhat more paternalistic streak to it in that it was meant in part to "civilize" those it conquered. But it was still driven by a clear sense of cultural, societal, and racial superiority.