On the surface, one would parse the incantation as something like “pesky pixie, pester no(t) me”, which is basically what the Harry Potter Wiki proposes. That’s fine, as far as it goes, but I have a bit more to add.
First, pesky — what’s the etymology of the word? According to the OED, it’s a mid-18th century U.S. colloquial word, “origin uncertain”, “conjectured to be an alteration of *pesty […]”. Most etymological dictionaries agree — e.g., Ernest Weekley: “[U.S.]. App[arently] from pest”. Alternatively, some suggest a source in the Irish Gaelic peasgach “troublesome”, from the noun peasg “impediment” . Whether the word is really of American origin may be debatable. Even if we ruled out the Irish source, Joseph Wright has noted roughly contemporary examples from Scots, Yorkshire, Oxford, and other dialects, and Eric Partridge has found evidence of a possible origin in Essex. Anyway, this is beside the point. What is the point? It’s that we’re not sure of the origin of the word pesky.
Well, here’s a theory I’ve never seen before — could it be related to pixie? It’s a longshot — especially if the word really did originate in America — but a case can be made. Let me offer this longish excerpt from Walter Skeat’s Notes on English Etymology:
Pixy. The Devonshire pixies, or fairies, are well known; in Cornwall the form is not pixy, but pisky, which I believe to be older. I once thought that pixy might be connected with puck, [… but t]here can be little doubt that the word is really Scandinavian; for there is no reason against the introduction of Scandinavian words into a county such as Devonshire, which is easily reached by sea. At any rate, it is well worth notice that the very word, with the same sense, is in use in Swedish dialects, particularly in South Sweden[, … including] the form pysk, more commonly pyske, pjyske, pjäske, pjöske, a little goblin [… etc.]. Now Rowling’s pixies are indeed from Cornwall, so they really ought to be called *piskies, shouldn’t they? And given this form, together with the “plaguey” nature of elfs and fairies in English folklore (cf. elf-shot, elf-child, etc.), it seems not altogether unreasonable to suppose pesky is not an alteration of *pesty, but of pisky.
Even if the words are not actually related — and they probably aren’t; after all, pest and pester are not (though you’d think they would be) — this view still informs the reading of Lockhart’s spell. The metathesis between the s and k sounds in both words is part of the amusing sound-play of the spell, along with the similar sounding pester. This particular transposition of sounds is quite common in the history of the English language (cp. Modern English ask, but Old English acsian, and in some dialects of Modern English, ax.).
See what happens when you let a few pixies out?! Now, I’ll ask you to just nip the rest of them back into their cage. :)
 Since this is not the mainstream view, I’ll offer two sources. Mackay, Charles. The Gaelic Etymology of the Languages of Western Europe. London: N. Trübner & Co., 1877, p. 323; and Hotten, John Camden. The Slang Dictionary. London: John Camden Hotten, 1865, p. 199.
 Skeat, Walter. Notes on English Etymology. Oxford: Clarendon Press, 1901, p. 218.