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.

As I understand your question and the comments, you are

• a citizen of Ukraine and/or Russia (having accepted a Russian passport during the "passportization" campaigns in occupied parts of Ukraine),
• currently in Poland,
• who has requested a visa for the US from Poland, and has been denied (possibly because of the response to a criminal record check), and
• who now wants a visa for the stated purpose of asking for asylum in the US.

The way Western governments interpret things, refugees are expected to stay in the first safe country they reach and to apply for asylum there. (As Relaxed points out in the comments, this is a self-serving interpretation which keeps them from having to accept many refugees. Ukraine was not a typical case because it is next to the EU. Refugees from other places need expensive smugglers and risky boat journeys.)

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.