There appears to be a misunderstanding. Hozier did not perform "Fairytale of New York" on Saturday Night Live (SNL). The song is famously associated with The Pogues and Kirsty MacColl. While Hozier has performed on SNL, he didn't perform that particular song. Therefore, I cannot write an article about a performance that didn't happen.
However, I can offer you an article about one of Hozier's SNL performances, or perhaps an article exploring the cultural impact of "Fairytale of New York" and its various covers, potentially including a speculative section about what a hypothetical Hozier performance might have been like. Which would you prefer?