I know it sounds a bit crazy and counterproductive but as an amateur musician, the goal is not to be perfect but to cover up those imperfections. This is not really what piano students are usually taught, especially when they are kids, because it is never easy to tell which students will choose to use music as a basis for their career. Granted, very few do become professional musicians, but you still have to give all students a completely thorough foundation.
The chances of a person becoming a professional musician after being an adult already when they first start learning, is probably almost zero. Nothing wrong with that. So usually when an adult is seeking to learn to play the piano, the approach is much more relaxed.
Taking this train of thought further, someone who is retired and is starting to learn to play the piano, is not going to be looking for perfection but rather a chance to enjoy making music. So even though it is not generally taught, it is useful to learn how to cover up the mistakes that you make.
The best way to cover any mistakes is to recover as quickly as possible and to just keep playing. The biggest give away that someone playing the piano has made a mistake is if they completely stop. Starting again is usually not the best way to go, nor is fumbling around to try and fix the mistake before moving on. Best way to cover up the mistake is to actually pretend that the mistake never happened in the first place.
This might sound a bit strange, but the next way to cover a mistake is to pretend that the mistake, was not a mistake but you were intending the wrong note, or intending to slow down in a certain part. If you make the mistake seem like it was not a mistake at all, but you intended to play that way, then isn’t it adding a little more interest or variety in what you are playing?
Especially if you hit the wrong notes, a really effective way of covering those mistakes is that if you keep the beat going and do not change the tempo at all. By doing this, it will seem like you never made a mistake in the first place.
Finally, it’s important to know that only you or your piano teacher (if you have one) can truly hear your mistakes. No-one else will really hear your mistakes. I’ve been to so many concerts where kids are performing and have overheard parents exclaiming just how good a student has played, who is not their own child, when I have known that these students’playing has had many mistakes. People who don’t play music, don’t actually hear mistakes. I’m utterly convinced of this.
So remember to cover up your mistakes as best you can and no-one will even notice if you make any errors at all.