Warren F. Broderick found five poems signed "H" in the Democratic Press and Lansingburgh Advertiser, published by that newspaper in 1838 and 1839. Broderick's article announcing the discovery ("Melville's First Five Poems?," Extracts 92, March 1993) is available in the online archives of Melville Society Extracts. Since Herman Melville did publish two "Fragments from a Writing Desk" sketches in the Democratic Press in 1839, it has seemed reasonable to conjecture that H. Melville may have submitted these "H" poems to his hometown newspaper.
But the first of the five "H" poems, "Pity's Tear," is definitely not by Melville. A search in Google Books reveals the rightful author as Mary Darby Robinson (1758-1800). Robinson's poem Pity's Tear shows up in volume two of The Poetical Works of the Late Mrs. Mary Robinson, 3 vols. (London, 1806) at 254-255. Lansingburgh's "H" could have found Robinson's poem reprinted in the posthumous London collection, or in an American newspaper or literary magazine, for example The Rural Visiter vol. 1 (Burlington, New Jersey) where "Pity's Tear" appeared without attribution in the August 13, 1811 issue.
Update 07/20/2018: "Pity's Tear" appears over the signature of "Sappho" in The Lady's Monthly Museum for September 1801.