Hey, Lena.
This is a tricky one, not least because your author was Ukrainian. That means any English rendering of his name is going to be phonetic - and therefore open to interpretation. You might find him referred to as
Bahryani or
Bariani or any number of permutations.
That said, there is a relatively large amount of attention given to his book
Tyhrolovy. The main English translation is called
The Hunter And The Hunted (1957, New York). To be honest, though, the story - about an poor family coping with the harsh Russian outdoors - doesn't sound very much like your description.
Anyway, even if it is right (and I don't think it can be, really) then the odds of it being out of print are pretty high.
Sorry I can't offer more help. You could try a
google using "Sad Hetsymans'kyi" as a search term - any sites it throws up with probably be non-English, but I assume that won't be a problem for you?
Best of luck.
Tobytook