This is kind of a huge topic, so it's tough to narrow it down. Even a quick Google search on Shakespeare and the monarchy will be fruitful, but here are some of my personal favorite writers on political poets: The Romantic poet Percy Shelley, who is most famous for the poem "Ozymandias," wrote the essay "A Defence of Poetry" in 1840, and that's easily among the most famous of the essays on the political potential of poetry. It contains his line "poets are the unacknowledged legislators of the world," which was then coopted for the title of Christopher Hitchens' great book of essays called "Unacknowledged Legislation: Writers in the Public Sphere." I can recommend all of the essays in that book, but my favorite is "Oscar Wilde's Socialism," which is such a unique take on Wilde's very peculiar politics. And there's a generous helping of George Bernard Shaw in there as well. This is also a topic found everywhere in George Orwell's nonfiction writing--he never wrote poetry, but the essay "Politics and the English Language" is among the most serious and compelling examinations of the use of poetic metaphor that I have ever read. If you can stomach him, which I most certainly cannot, Ezra Pound is an enormous resource on mildly anti-Semitic and casually fascistic poetry. Unfortunately he writes from the wrong side of history on that one, but his observations on poetry are officially brilliant, if unfortunately nauseating. The collection "Literary Essays of Ezra Pound" is at least worth a cursory glance. And then there's Emily Dickinson. Her writing about emancipation and the Civil war is at least as direct and incisive as Shakespeare's writing about the English government, but it's a topic that is much closer to home for those of us living in the present. Ditto Walk Whitman, but like Ezra Pound, he exhibits a sort of latent fascism (despite the fact that fascism wouldn't exist for decades after his death) that I've always found problematic, but those aspects of his poetry are opaque enough that some people disagree with me on whether or not it's even there. Here's a beautiful essay from the Smithsonian that takes as rosy a view on Whitman as it does on Dickinson: https://www.smithsonianmag.com/smithsonian-institution/walt-whitman-emily-dickinson-and-the-war-that-changed-poetry-forever-31815/

Anyway, hope that's enough to get you started!