Mistakes in playing piano songs can be fixed by having a feel for each of the songs as a single whole entity and an understanding of the composer's spirit and style
In practice I have learned to accept mistakes. There was a time when I set out to play without mistakes in every practice session, thinking such a goal would improve my technique faster. Instead the outcome was frustration, stress, and a devastated morale.
Music making is a process that could not be forced. By practicing continuously and regularly, making the necessary corrections here and there, the mistakes gradually disappear. When they do not, I had to accept that; today is not the day and needed to spend more time on the problematic bar or phrase. Let them be for now, returning to them tomorrow.
This approach has helped me gain a sense for the composition as a single whole entity. Because even with mistakes, I played through from the start of a phrase to its end, thus able to hear and feel the content it expresses. Work on correcting the mistakes is done by practicing the phrase repeatedly had the effect of clarifying further its artistic image. As a result, I became more intimate with the phrase and could express it better.
Mistakes most common usually fall into two categories; tempo and rhythm. Both are direct consequence of not acquiring a sense for the composition as a single whole entity and a clear artistic image.
Which could be remedied by staying with one composer for an extended period, learning to play many of his compositions.
By doing so you get an understanding of his spirit and style. I listened to different versions of Canon in D which included the original by Pachelbel, an arrangement by George Winston, performed on the piano and violin. When I was ready to learn to play it on the piano the artistic image was vivid in my mind. I was able to guide myself through the composition in reproducing the rhythm as accurately close to the ideal as possible.
Mistake in tone could lead to mistake in rhythm. Because the piano student tends to produce a soft tone when executing a slow tempo and a loud tone when executing a fast tempo. Especially inexperienced students. A semiquaver could be played as a quaver by mistake due to this tendency thus distorting the rhythm. The phrase shown in Image 1 below is an example of such a mistake I have also made in the past.
As shown in Image 1, a phrase does not necessarily slow down when it is soft and accelerate when it is fast. Regardless of tempo, tonal quality is maintained. To avoid falling into this self-inflicted trap, heighten your alertness when confronted by such a phrase. I resorted to deliberately practicing the phrase soft and moderately fast (tenuto), then switching to playing it soft and slow (albeit the wrong way to play it, only as an exercise) to develop control in tonal quality while changing tempo. Being able to maintain tonal quality at will while changing tempo reduces the risk of committing the error in tone production.
On the mechanical side of things, touch affects the tonal quality. Therefore, strong fingers that are also flexible needed to be developed. An exercise I had been using for years is to position the left hand’s 5th finger on C but not pressing on it. While pressing the 4th finger on D and holding the key down, play the rest of the notes.
Somewhat mirroring the left hand, position the right hand’s thumb on C but not pressing on it. While pressing the 2nd finger on D and holding the key down, play the rest of the notes.
Increase the level of difficulty to gain more flexibility in the fingers of the left hand by playing C with the 5th and D with the 4th fingers one at a time while fingers 1,2, and 3 stay pressed on the keys of E. F, and G.
With the right hand, play E with the 3rd and F with the 4th fingers while fingers 1,2, and 5 stay pressed on the keys of C, D, and G.