Poem
late 15th century: from French poème or Latin poema, from Greek poēma, early variant of poiēma ‘fiction, poem’, from poiein ‘create’.
wiktionary
From Middle French poème, from Latin poēma, from Ancient Greek ποίημα(poíēma), from ποιέω(poiéō, “I make”). Displaced native Old English lēoþ.
etymonline
poem (n.)
1540s, "written composition in metrical form, a composition arranged in verses or measures" (replacing poesy in this sense), from French poème (14c.), from Latin poema "composition in verse, poetry," from Greek poēma "fiction, poetical work," literally "thing made or created," early variant of poiēma, from poein, poiein, "to make or compose" (see poet).
From 1580s as "written composition, whether in verse or not, characterized by imaginative beauty ion thought or language." Spelling pome, representing an ignorant pronunciation, is attested from 1856.