Hi, I would consider two things.. first is what Nihilist has already said about slowing down. I have a phrase -
'Give yourself time to get it right' A lot of the time people don't give their brain enough chance to correctly perform the task that is new to it, so they trip up and go back to the start and then trip up over the same bit. If you're tripping up consistently in the same area(s) then it's definitely a case of isolating the problem area and giving yourself the time to play relaxed, focused repetitions.
Regards making mistakes whilst playing a run through of a piece - don't stop. What you're doing is actually programming yourself to stop when you play a wrong note, when in a playing situation you need to be able to ignore it and carry on. You need to train your mind to ignore mistakes and still maintain the rhythm of what you're doing. So, if you're committing to playing a run through of an entire piece, do it. If you stop every time you make a mistake you won't get the end of the piece for ages.
If you approach things two ways you should be able to tackle both issues:
1st approach - leave mistake correction to focused practise time where you isolate and repeat problem areas
2nd approach - when playing an entire piece all the way through, play it all the way through - don't stop if you mess up because you can't do that live. Well, you can but you wouldn't want to
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