I'll give my opinion even though I am new to the forum. I naturally agree completely with Paolo and I find the sentence said at the beginning to be absolutely correct.
If you, Daniele, study a piece and during this study (variations and metronome bursts are an excellent thing, by the way) you don't work well, meaning you allow yourself mistakes, these wrong traces remain in memory and the brain is worse than a computer, damn it! Let me explain better. If in a piece the hand instinctively—whether out of laziness, by chance, or for performance practicality—wants to play one note instead of another that might be uncomfortable, then that instinctive gesture, which is therefore easily absorbed by your brain, will with difficulty reverse all those mental gestures that you performed correctly while reading the score.
When tackling a piece, it is generally read in parts, reaching a precise point, but this should not be underestimated since that virgin moment is fundamental; it is at that moment that you must not allow yourself (or rather, you must not allow your hand to do whatever it wants) but with the help of the brain, you must concentrate (perhaps with great effort) and try to say all the right notes in the right place. You must not just repeat the piece or the set of notes that form it, but mechanically, because it is precisely that type of mechanical study that struggles to take hold in the memory. Let's say that technique and study should be experienced as joy and not as tedium, and all the difficult and technical parts must be studied in function of expression and not coldly. For example, have you never noticed how pieces with slow tempos stay in your head more than those with fast tempos? Because a slow tempo has time to settle in an orderly and rhythmic way within our brain, and since it is harder to make mistakes on an "adagio" compared to a "presto," this adagio will be remembered better, in addition to the fact that the brain has much more time to choose the notes it needs to say at the moment of execution. Anyway, every individual has different aspects to resolve, but what you are saying is quite common and is happening right now even to me when I underestimate certain situations.
Regarding studying hands separately, I think it is right, but I believe one should also do a lot of sight-reading practice with both hands together.
Sorry for the somewhat long response, but I think that by talking about how I resolved some hiccups, maybe someone might benefit, right? Naturally, if there are inaccuracies or bad advice, I hope our supervisors intervene, who... are truly well-prepared. Bye and thanks for this forum.