How can I properly clean fingerprints on a new MacBook Pro M4 screen?

I got my new MacBook Pro M4 recently and while opening it, I got a few tiny fingerprints on the screen's side. I used a microfiber cloth lint free (or at least I hope, it's the same one I use for my glasses). I dampened it slightly and moved it across the screen thinking it would clean but it just spread that cloudy look! It is like when wiping ketchup of a counter, it just spreads! Now almost half the left side of my screen top to bottom looks cloudy. When I use bright websites, you can't notice it but in dark scenarios like Netflix. You can clearly see the clouds. Have I done something wrong? How do I fix it?

[Re-Titled by Moderator]