How could I go about writing four pages about the essay "Mother Tongue" by Amy Tan, citing two or more different authors to support my claims?

Since you did not say what claims your essay needs to make and support, I will assume that is up to you. An interesting approach to "Mother Tongue" by Amy Tan would be to compare it to essays from previous centuries by authors who wrote about writing. You could claim that Tan is maintaining and updating a tradition of authors who analyze their craft and the way they use language. Two sources I recommend are "Silly Novels by Lady Novelists" by George Eliot (Marian Evans), written in 1856, and "Politics and the English Language," by George Orwell (Eric Blair), written in 1946. 

While pointing out first that neither Eliot nor Orwell concerned themselves with writing for an audience whose first language was not English, you can nevertheless assert that both Eliot from the 19th century and Orwell from the mid-20th century—like Tan in the late 20th century—abhor snobbery in writing and embrace honesty.

Eliot, for example, has this to say, speaking of the "woman of true culture":

In conversation she is the least formidable of women, because she understands you, without wanting to make you aware that you can’t understand her. She does not give you information, which is the raw material of culture—she gives you sympathy, which is its subtlest essence. (p. 196 on link below)

This sounds very close to the conclusion Tan eventually arrived at when she realized that understanding, not grammar, was important in reaching her Chinese American readers, including her mother. Eliot also mocks writers who use elevated diction and whose writing is marked by "a careful avoidance of such cheap phraseology as can be heard every day" (187). Tan learned that using everyday speech in her dialogue, even with its grammatical inaccuracies, was much more engaging for her readers.

Although Orwell's essay advocates for improvements in the English language, he doesn't mean a blind adherence to grammar rules. In fact, he states the defense of the English language "has nothing to do with correct grammar and syntax, which are of no importance so long as one makes one's meaning clear." In this he would support Tan's use of "broken" English to communicate with her audience. After spending most of his essay mocking the snobbery of certain ways of speaking and writing in English, a snobbery which Tan admits she practiced in her early days, Orwell states, "If you simplify your English, you are freed from the worst follies of orthodoxy." This is certainly what Tan learned as she embraced the simple beauty of her "Mother tongue."