As I understand your question, you are a citizen of the Ukraine and you currently live in Poland.
The international treaties for political refugees say that you can flee from the place where they are persecuted to the next safe place on their escape route, apply for asylum there, and have their request evaluated. So you would have to explain that you are persecuted in Poland, and the US does not generally believe that Ukrainians are persecuted in Poland. While it is possible that you are an exception, it would be difficult to convince the American officials.
Since the US now has your first attempt in their database, any future application will become even more difficult.