I had speculated that maybe the family comes from Hanow in Eastern Poland, near the border to Belarus and Ukrania
Hanów, gmina (municipality) Dębowa Kłoda, Parczew district, Lublin Voivodship. It lies 35 km West from Włodawa (on the Bug River, close to the borders with Belarus and Ukraine).
Hanów is a very young settlement. The first mention comes only from 1921's first census of the Second Republic. The village was established in the late nineteenth and early twentieth century on the part of the village Nietiahy. The name of the village comes from the name Hanna. In 1921, the Hanna's colony of 64 people lived in 14 houses.
In that quote they used the word "colony". In Poland, it is usually applied to German settlements.
Just 20 km North of Włodawa, there is village Hanna, gmina Hanna, Włodawa district, Lublin Voivodship.
For what I understand, both Hanów and Hanna could not serve as your family nest.
where, by the way, the name would be spelt with an "-sky" at the end.
This does not compute historically. This is true that -ski ending Ukrainian surnames are indeed being transliterated to Latin - or to be more exact to English - as -sky. However, this is just a modern rule. Recall that the Ukrainian language and alphabet was still undergoing transformation and standardization as late as 1927 and 1930s. Early documents, mentioning Polish and Rusyn gentry were written in two languages: Polish (Latin) and Rusyn (Cyrillic). Nobody transliterated from Ukrainian to English then. :-)
So the -ski names remained -ski in Polish version and -ський (or something like that) in Rusyn. In early 15th c. Rusyn gentry, at the Eastern part of Red Rus (Grody Czerwienskie), near Lwów and Żydaczów, were still using toponymic forms: Jursza of Chodorowstaw, Stańko of Dawidow (in 1410 he was still known as Ostaszko of Dawidow), Dmytro Lahodowski, Martyn Kalenyk of Podhajec, Michno and Paszko of Borszczow, Juryj of Malczyce, Senko Halka of Iljaszow, Senko of Nahorce, Olechno, Marko and Lenko of Drohoszow, Petro Wolczko of Kolodence, Stecko, Onyszko i Stecko-Ilko of Czerkasy, Dmytro and Jacko of Didoszyce (Diduszyckis), Jacko of Roznitow, Andrejko of Swaryczow, Iwaszko of Duliby, Iwan of Koszawa, Oleksa and Luczko of Witwice, Danko, Myka and Senko of Balice, Jacko of Nowosielice.
The Western part of Red Rus were already using -icz, -cki and -ski forms, such as:
Andrijko and Hryćko Bybelski, Hryćko Kierdejowicz, Hlib Diadkowicz, Wolczko Presłuzycz, Danyło Zaderewicki, Kostko Solneczkowicz, Kostko judge of Przemyśl, Jacko judge of Sanok, Waśko Teptiukowicz, Mychajło Procowicz, Drahut Wołoch, Chodko Czemer, Juryj i Waśko Moszonczycz, Wołczko Kuźmicz, Mychajło Senkowicz, Iwan Danslawicz, Oleszko Hrudkowicz, Waśko Czortkowicz.
So why the -sky form appears in some German or Polish names. It is an effect of Germanization, undertaken at various times, that the -ski, -cki suffixes had been replaced by -sky and -tzky. For example: Czarnowsky, Lubawsky, Naczyńsky were the names common to Northern Poland ( not Ukraine) - Pomerania, Warmia, Kashuby (Royal Prussia), respectively. These areas - plus Upper and Lower Silesia and Lubusz region - are the areas where I would look for Hannowsky variation of Hanowski, not in Ukraine.
But where the -sky suffix came from? From Bohemia, from our Western Slavic cousins. Germans had been exposed to Czech names for much longer period than to Polish names, and accepted the Czech-like suffixes -sky as more natural than -ski.
Czechs spell the -ski ending names with -ský, where ý is known as dlouhé ypsilon (long y). An acute accent over a vowel signifies a long sound. Czech -ský sounds exactly like Polish -ski, while their pronunciation of -ski sounds like -sky in Polish. [This and other such false friends are jokingly referred to as Czech errors in Polish]
For example, Sikorski pronounced by Czechs sounds to Polish ear like Sikursky, while Šikórský as Sikoorski - almost right so as Polish Sikorski.
(With some approximation, since in Polish S becomes palatalized when followed by "i". Hence I replaced Si by Ši to approximate that soft sound).
This is all due to the fact that both languages, though close cousins, have developed their spelling and pronunciation quite differently over the ages.