Overall, when it comes to geopolitical enemies of the United States in particular, it would not be a bad idea to treat your current understanding with extreme skepticism until you’ve investigated counter-sources as well. That doesn’t mean the US always lies, in fact it frequently tells mostly the truth, but will distory either the quality or quantity of an event.
This is a decent overview of the background that led up to the events of 1956, and this is a decent overview of the darker side, where the lynchings happened. Content Warning: lynched corpses. Here is a source on MI6 training and arming the counterrevolutionaries. Those 3 articles give only the briefest overview of the events, but don’t do the real buildup to them, their complexities, what the people actually supported, or the real character in any depth. If you want to actually take a deep dive, these are additional sources:
The History of the Working Class Movement in Hungary
1956 Counter-Revolution in Hungary
Others can offer more sources.
Overall, when it comes to geopolitical enemies of the United States in particular, it would not be a bad idea to treat your current understanding with extreme skepticism until you’ve investigated counter-sources as well. That doesn’t mean the US always lies, in fact it frequently tells mostly the truth, but will distory either the quality or quantity of an event.