The first time I saw this word, I was struck by what an awkward-looking thing it was, and by my uncertainty as to how to pronounce it. There is persistent confusion, in fact, as to the latter; the version with stress on the second syllable – “Hey, Jiminy! You have hegemony over the crickets!” – is perhaps the more “proper” (preferred by those who arguably have hegemony over English usage), but the version with stress on the first – “The banking system is suffering from the hegemony of hedge money” – is common enough. One thing to nail down at any rate is that the g sounds like a j, even though the Greek source, hegemon, “leader,” had a [g]. We used to make that change as a matter of course in English, and we also represented any j sound before e or i by a g as a rule. That this has changed for imported words is illustrated by the fact that we now generally say [g] in Gibran and Genghis even though these relics of an earlier era of transliteration would have been Jibran and Jingiz (or Chingiz) had they been set today.
You’re probably most likely to meet hegemony in an academic context, likely in cultural studies or perhaps history. Those who criticize hegemonies are themselves typically among the oligarchs of the academic hegemony over theoretical discourse. You would get the sense that hegemony means “this group is hogging all the toys to themselves,” but really it refers tout court to leadership or preeminence, and not necessarily to a quasi-autocratic dominance. The academic hegemony over discourse on hegemony is illustrated by the words that appear often in the neighbourhood (within a few words) of hegemony: Gramsci (Antonio Gramsci, Italian Communist leader and philosopher imprisoned by Mussolini, whose writings on hegemony inform much of the discourse on it), bourgeois, masculine, ideological, and even hegemony used again nearby, as well as dominance, cultural, regional, global, and a whole lot more of the same ilk, including of course American.