。
She read him a poem.
Literal
She [topic-は] him [to-に] one-piece [genitive-の] poem [object-を] read-and-did-for-him.
一編 (いっぺん) is the counter for poems, articles, novels, films — units of literary or artistic work. 一編の詩 = 'one poem.' The benefactive ~てやる adds 'did this for him' nuance, with やる (vs. あげる) signaling close or familiar relations. 詩 ('poem, poetry') is one of the few common single-kanji nouns in modern Japanese.