In Polish, HARASIUK is pronounced roughly "hah-RAH-shoo." HARASUK would sound more like "hah-RAH-sook." As I'll explain below, I suspect HARASUK is a misspelling or a variant, and HARASIUK is the version to concentrate on.
2002 data on surnames borne by Polish citizens suggests HARASIUK is the form you're more likely to run into. That data showed no one in Poland going by HARASUK, but 521 Polish citizens named HARASIUK. You can see the data here, along with a color map that illustrates it nicely:http://www.moikrewni.pl/mapa/kompletny/harasiuk.html
The name is most common in eastern Poland, near the border with Belarus and Ukraine. That is not surprising, as I'll discuss in a moment. On that map, position your cursor over a powiat (county or district) to see its name; that's how you tell which is which. HARASIUK shows up most often in the powiaty of Bilgoraj (73), Nisko (66), Sokolka (41), and Tomaszow Lubelski (26). This means it's most common down in southeastern Poland, in the around area the city of Lublin, near the border with Ukraine. But it also shows up in Sokolka powiat, up in northeastern Poland, near the border with Belarus.
The late Polish name expert Prof. Kazimierz Rymut mentions this name in his book _Nazwiska Polakow_ [The Surnames of Poles]. He lists it among numerous surnames that derive from a first name used in various different forms by Slavs, including Gerasim, Harasym, Harasim, and so on. It comes from Greek Gerasimos, from a root meaning "honor, privilege," and probably was meant as a name of good omen, in hopes a child so named would grow up to be honored and privileged. (There's a little more info on it here: http://www.behindthename.com/name/gerasimos
). The name came from Greek into use primarily by Eastern Slavs (Belarusians, Russians, and Ukrainians) -- it's not as common among Poles as it is among them.
The fact that the name shows up in Poland most often near the eastern borders suggests it originated in Belarus or Ukraine, in lands formerly ruled by Poland. In those regions, the initial G would tend to become an H sound, Gerasim- to Herasim-, and the first vowel would tend to become an A. Then that could be shortened to Haras (HAH-rahss) or Haras' (HAH-rahsh) or Harasz (HAH-rahsh), kind of like the way we shorten "Gerald" to "Jerry" or "Harold" to "Harry." If you add the diminutive suffix -iuk to Haras, you get HARASIUK, which means "son of Haras"; or you can add plain -uk to Haras' and the accented S becomes -SI-, also yielding HARASIUK. HARASIUK or HARASUK, therefore, would be names of Eastern Slavic origin meaning "son of Haras/Haras'," which started out as a nickname for Harasim.
The name probably s