How does the image "The Two Great Missioners of Civilization" relate to American perceptions about the manifest destiny of the United States and about the U.S.'s role as a global power around the turn of the twentieth century?

In answering a question like this one, you need to think about a few things: the nature of the source (who created it, and for what purpose?), its context (what was going on at the time?), and the content of the source (what does it show, and what attitudes are represented?).
"The Two Great Missioners of Civilization" is a political cartoon by Victor Gillam that was published in Judge, an American satirical magazine, in 1898. It shows the United States, in the form of Uncle Sam, striding through a destroyed landscape and scattering pieces of paper which say such things as "liberation" and "savior." Uncle Sam is accompanied by another stock comic character: John Bull, the personification of Great Britain. In the destroyed landscape surrounding the two characters (who appear to be bringing a new railway into the country along with their national flags), native people are being trampled in the undergrowth. Symbolically, this means that the promises of liberation are falling into the grass rather than reaching the people they are intended for.
This cartoon, then, is a stark criticism of the idea of the United States's "manifest destiny" as the rightful savior of the world, a popular ideology in this period. When this cartoon was published, the United States's position in the world was changing; in 1898, the Treaty of Paris marked the beginning of the United States's colonial power as it took possession of Puerto Rico and the Philippines (among others). The U.S. was more populous than other global powers, and had possession of a considerable military might. Many saw it as the destiny of the United States to go forth and Christianize, or "save" countries which now fell under their protection. This typified the idea of "manifest destiny": that the U.S. had not only the right, but the obligation to further colonize the world.
However, many argued against this approach for various reasons. For example, Andrew Carnegie said there was little point in reaching out into the Philippines to "make Americans," because these people, he argued, could never become Americans due to their racial origins.
Mark Twain also eschewed this idea; he argued that America's imperialistic attitudes flew in the face of its own revolutionary origins. The Unites States, he argued, had become independent in defiance of the control of Great Britain, a control the U.S. then started to imitate through its imperialistic actions. The presence of John Bull in the cartoon reflects this, as it suggests that America was becoming like the power it once escaped.
https://npg.si.edu/object/npg_AD_NPG.77.13