I can't address enough how important singing is for piano (or any other musical instruments). Some of my students (especially beginners & intermediate students) don't want to sing even if I require them to sing while playing some passage. They say, "Oh, I'm shy.," "No, I don't have to sing. I'm OK without singing.," or "I'm a bad singer." Well, it doesn't matter what you think about your singing. Even if you are a terrible singer, even if you can't sing in pitch, I encourage all piano learners to sing what they are practicing. This is important, especially if you are having trouble with a certain phrase. Oftentimes, the reason why you keep making a mistake the same way over and over is that you are not hearing the passage or phrase properly in your mind in advance. That means your intention is not strong enough. You may say, "I don't know why my fingers don't go to the right notes the way they supposed to go." Well, once you hear it in advance, your intention is going to be very strong. Then, your fingers will be forced to go to the right notes, or do the right thing. There is a push behind it. You can't just try to do everything with your fingers mechanically. So, that means you need to sing. Singing will strengthen that push, that intention. When you sing, you have to use your breath, your voice. Naturally, your intention is going to be strong.
Piano players (especially beginners & intermediate players) don't use their voice (or breath) enough. Once you incorporate singing into your routine, your playing skills will dramatically improve. I highly recommend all of you to take singing lessons. If you can't do that, how about joining a local chorus group. Or, just try to use your breath and singing into your piano practice all the time. For example, if you are practicing just a melody line with your right hand, sing or hum along with your playing. You don't have to sing words. You can just hum, "La, La, La," or "Tah, Tah, Tah," "Nah, Nah, Nah," whatever...
When students go to music school or conservatory, all students no matter what their instruments are, are required to take a singing class because it is extremely important. The first musical instrument is VOICE. It's most organic form of musical tool.
Players for other instruments such as brass & woodwind instruments are required to use their breath. So, they are accustomed to that, and they can easily be ONE with their instruments that they are playing. Piano learners have a disadvantage because it's easy to just create noise by just pushing the keys with fingers. But that's not music. Music needs to come from your breath. You use the breath and express yourself onto the piano. (I'm not saying that you should blow air onto the keys.)
Even if you don't believe me, please try singing the passages & musical phrases while playing those on the piano. Experiment with that. I'm sure you'll find it very helpful.