Just curious, what does it really take for evidence to be thrown out of a criminal trial? Re: Luigi Mangione

I recently saw online that Luigi is in hearings to have evidence (his backpack with the murder weapon and manifesto) deemed inadmissible(?). I guess the officers didn't have a warrant yet when they searched the backpack, and the defense is claiming that the officers were already convinced he was the assassin and they were biased. I don't know how much of that is true, but it made me wonder about how this kind of thing works in real life.

Theoretically, how hard is it to get evidence (particularly something like a murder weapon) thrown out? For the evidence to be considered admissible, is it enough to have had probable cause, even without a warrant? And what constitutes probable cause in this sort of situation? Is it enough to say that someone looks like a wanted person who is considered armed and dangerous?

Without knowing the details, is it probable that the evidence could get thrown out in this situation, or is it just standard procedure for the defense when there's no warrant despite it being unlikely to succeed. It seems like the kind of thing that the internet is making it out to be a bigger deal than it actually is because we laypeople all went to ~Legally Blonde Law School~ and have no idea how criminal law actually works.