In a way, "Mrs Faust" is one long allusion. It satirizes the Faust legend, a tale from German folklore, as well as other famous literary works inspired by that legend.
A detailed history of the Faust legend can be found here, as well as in the Literary Context section of this guide. The tale got started in the 16th century, when the anonymously authored Faustbuch (1587) combined older stories about wizards and necromancers with rumors about a historical Faust (a scholar and magician) who had died a few decades earlier. Christopher Marlowe's play The Tragical History of Doctor Faustus (c. 1592, published 1604) then popularized the tale in English. Over two centuries later, J. W. Goethe wrote the two-part, German-language play Faust (1808, 1832), considered a classic of world literature. In all of these versions, Faust is a scholar/magician who makes a pact with the Devil (or his representative, the demon Mephistopheles), offering his immortal soul in exchange for knowledge, pleasure, wealth, and/or power.
Duffy's poem adapts this tale into a turn-of-the-21st century context and adds the character of "Mrs Faust," not present in earlier versions. Her version imagines the Fausts as a rich, trendy, greedy couple determined to acquire the best life has to offer (from houses to yachts to "toys" that were new in the 1990s, such as "mobile phones" and "Internet"-connected home "computers").
Along with the general story, she also modernizes some specific details from previous versions. For example, in Marlowe's and Goethe's versions, Faust magically conjures Helen of Troy—the most beautiful woman ever, according to Greek myth—from the underworld and falls in love with her. (Goethe's Faust has a child with her.) By contrast, Duffy's Faust is "pleasured / by a virtual Helen of Troy"—apparently some sort of pornographic digital simulation. Marlowe's Doctor Faustus also peppers Mephistopheles with questions about the solar system and universe; in the poem, this curiosity about astronomy is reflected in Faust's voyages to the "moon" and "Sun" (lines 61-63).
It's also significant that Duffy's Faust, as in the Marlowe play and other older adaptations, goes to hell in the end—unlike Goethe's more sympathetic Faust, who is spared and goes to heaven. Anyone adapting the story in the modern day has a choice of classic endings, and Duffy chooses to make her Faust as villainous as possible.
Duffy alludes to other literary works throughout the poem as well. The line "spun gold from hay," for example, references the fairy tale Rumplestiltskin, in which the titular character weaves straw into gold. Rumpelstiltskin is able to perform the task only with the help of magic. Here, the implication is that Mrs Faust's money—like magic—makes the impossible possible.
"Bo Beep" in line 72 alludes to the famous English nursery rhyme "Little Bo-Peep." The implication is that Faust is cruising Internet dating sites for innocent young women to seduce (and perhaps bring into his sheep-cloning business). This marks another parallel with J. W. Goethe's version of the Faust story, in which Faust seduces a virtuous young woman named Gretchen.