and cursed the bread.

Obviously, though, Richard Corey did not see himself as others saw him, for 

Richard Cory, one calm summer night,
Went home and put a bullet through his head. 

The first theme you could explore in an argumentative or persuasive essay is that we should not judge a person solely by appearances. By all outward measures, Richard Corey has everything anyone can want and people envied him; despite that, Richard Corey was not happy. In fact, he was so miserable and discontent with his life that he committed suicide. 

A second possible theme is that wealth and position and even personal magnetism cannot ensure happiness. Clearly Richard Corey had everything that others wanted to have and be, but none of it made him happy, In fact, he was so unhappy with his life that he wanted to end it.

So, you could use this poem to make the case that it is not wise (or accurate) to judge people based only on what is on the outside: appearance, money, talents, and other external factors. I'm confident you can think of plenty of reasons and examples to prove this point.

You could also make the argument that possessing things (wealth, beauty, popularity) does not constitute happiness. Again, you can undoubtedly think of many examples to prove this point.

Both of these basic ideas would be terrific starting points for a persuasive or argumentative essay. To help you, I have attached an excellent eNotes site (linked below) on how to write an argumentative essay in nine easy steps. 

I like to share a thesis statement template, similar to this one, with my students when we discuss poetry: "[Title]" by [Author's Name] uses [poetic device], [poetic device], and [poetic device], to convey the idea that ________________________.

Therefore, if you were trying to write a thesis statement about this poem, you might say: "Richard Cory" by Edwin Arlington Robinson uses word choice, imagery, and irony to convey the idea that we ought not judge someone by their appearance. Robinson uses words like "crown," "imperially," and "king," bringing to mind royalty and suggesting the idea that Richard Cory seems like royalty to everyone else in town. He is said to "glitter" when he walks, a visual image that makes Cory seem special, even superhuman, and he "flutter[s] pulses" when he speaks, a kinesthetic image that helps us to feel the effect he has on others. All of these aspects of Cory lead others to think "that he was everything / To make [them] wish that [they] were in his place," and so neither they nor we, the readers, expect that he will go home on "one calm summer night" and "put a bullet through his head." This situational irony helps to create the poem's main idea, that we cannot look at another person and know them or understand them. Everyone thinks Cory's life is so enviable, that he has everything one might want, but this is clearly not the case. If it were, he would not take his own life.

I believe what you are asking is, "If this poem is making an argument, what argument does it make?" An obvious argument the poem makes is that money cannot buy happiness. Richard Cory is described as wealthy, well dressed, well educated, polite, and respected. The narrator states that Cory had everything to make the townspeople wish they were in his place. Some who are wealthy look down own those less fortunate than themselves, but this did not seem to be the case with Cory, for "he was always human when he talked." Yet something was missing in Cory's life, and we are never told what that was since the poem is narrated in the first person by someone who only observed Cory from a distance. We don't know what inner turmoil Cory suffered, whether he was clinically depressed, or whether he had a guilty secret that drove him to suicide. Whatever his motivation for taking his life, it was something that all his money was not sufficient to overcome. Even though he was "richer than a king," his life held torment. Those who were watching him from afar and wishing they were like him did not realize that in some ways they were better off than he was. The poem leaves us suspecting that the townspeople re-thought their envy and perhaps learned to appreciate more about the positive aspects of their own lives.