It's hard to describe in writing, but I'll try. There's no "h" sound, like the one where you just exhale, neither in Russian nor Ukrainian. The Russian "г" is hard, like "g" in "get". The Ukrainian equivalent to that is "ґ", but their "г" is something in between, closer to "kh". The fact that many English-language reports about the war do transliterate "г" as "h" in Ukrainian toponyms doesn't help, but it's as close as one could reasonably get to the real Ukrainian pronunciation.

There's so much no such sound that I've seen Hebrew and Arabic transcribed into Russian with the character "h" for that sound.

> Does this phenomenon have some name, as googling didn't turn out much?

Certainly not a formal one. Хэканье? Or maybe simply украинский акцент?