Not to belabor the point, but, in spite of received criticisms, it seems to me posthegemony has not been dealt with, it has only been swept under the carpet, quite conveniently. This is a bit frustrating. It is not that many ignore the term that bothers me–what bothers me is, rather, that many who know the term choose to ignore it when they write about issues and contexts that call for the deployment of the thought posthegemony was meant to elicit. It is more like censorship, really.
The way the term developed, in the context of discussions in the late 1990’s on subalternity, it meant to say (this covers, I think, all versions of it, including the most famous one by Beasley-Murray; but posthegemony is quite differentiated internally, so critiquing one or the other version of it is not the same as critiquing the whole thing) that hegemony does not and cannot exhaust the political field; that hegemony cannot and should not become the central focus of a (total) theory of the political (although of course there can be a theory of hegemony); that hegemony is not a necessary condition of politics; and, certainly, that hegemony is not a sufficient condition of politics.
More proximally, posthegemony came up as a term for what we thought was a necessary critique of Gramscianism (in the 1990s: the necessity of it has only increased exponentially); for a radical critique of “actually-existing” communist politics, even if by “actually existing” we refer for the most part, nowadays, to the politics of those who declare themselves communist in academic discussions; and for what we thought was a necessary critique of post-Marxism as it came to us from Ernesto Laclau and Chantal Mouffe and others.
The silencing is so loud that, naturally enough, those who would otherwise be inclined to get into discussions about it decide in advance it is too damned dangerous to use it. This is, ironically, what hegemony does to thought.
Infrapolitical Deconstruction (and Other Issues Related and Unrelated.) 
I don’t care for this term. We have discussed why elsewhere. You say it is not teleological, that whatever the ‘post’ indicates, it is not for taking literally. Ok. But, nevertheless, it bores me before my eyes reach the letter ‘h.’ More interesting (much more) is the notion that, however the would-be hegemony of thought presents itself–which always seems daunting, manifesting, oppressing–it is only fleeting, a not-so-not-really type of thing. Hegemony is an abstraction that runs at the speed of insecurity. Even critiquing the hegemonial, which when loud enough and popular enough itself becomes hegemonial, creates a duumvirate; but it can only play-act a fake competition for false titles of leadership. So why seek a claim on it at all? to call it out, to belittle it, to make fun of it (this is an admittedly good reason for keeping pothegemony, to make fun of hegemony, but it seems petty, especially a quarter of a century later), even though the term already wants a push away from hegemonic fantasy? Perhaps I simply tire of explanations, teachers, prophets, leaders and leadership. No more building, no more tearing down. A nomadic life, with a few settled spots to refill on food and drink (and those settled spots know death and finality like intimates, their entrenchment is a claim over longevity that no badlands saloon owner actually believes in). That is the thinking; that is where humanit-y-ies (used to) live, and where they, we, are returning: a primordial state of evolution–not moving, not developing–but changing along with and because of the micros, the bacteria of bacteria, that colonize, settle on, and finally expel (us/them). A tangible hegemony? Impossible. A posthegemony is rather about the intangibility of that which would like to seem hegemonic. But, still, I can’t help but giggle.
Alejandro, you are perhaps quite close to Jon Beasley-Murray’s position on this. In any case, a further point I would like to make: posthegemony addresses a certain number of problems that are clearly problems for the left today, as recent history has abundantly confirmed (in fact, it has done nothing but confirm it everywhere). And yet that same glorious left continues unabated saying the same old same old and refusing to register that their position is stale, rancid, paralyzed, and going nowhere. In fact, they even repeat the same jokes and seem to have no realization that nobody laughs any more. And they call that, I suppose, loyalty? To past incompetence? O wonder of wonders.