My phone records piano music quality very poorly. I am using the built-in microphone.
I am humbly asking whether you could suggest an app for me that could record high quality music for my playing technique and my voice too.
It's not the app, it's the phone's hardware that is limiting you. Your phone has a mic and an analog-to-digital converter inside of it that picks up the sound and converts it to digital for whatever app to record. Because your phone is small and engineered to be of use to ordinary people and not audio engineers, those hardware components are not very good relative to professional recording standards.
No matter what app you use, it will be getting the same low quality input from the same hardware. So the solution is to get some kind of external hardware that is of better quality. There are audio interfaces for phones and computers and there are standalone recorders that would all work.
You definitely will want a good external microphone.
There are several types available, from the usual sources.
I recommend you begin your search by looking for a good lapel mic to start with. They plug in to your regular 1/8" jack, just like the earbuds with microphones in them, but are of much greater quality.
They work well, and you would want to clip it on to the shirt of the singer and let it pick up the piano in the background.
Here is the one I use: Quality Lavalier Lapel Clip On Microphone
For about the same price you can go el-cheapo and try a small add-on microphone for your phone like this one: Olympus Noise Cancelling Microphone - I haven't tried this because I like to have a cord between my tablet and the microphone so I don't get rustling noises as I control the tablet.
Another option is to get one that can plug into the USB port. I have good success with one of those. I am using a tablet with a regular USB port so can't really address how well it would work on a phone. At the very least you would need an adapter that allows a full-size USB plug to go into your phone. Not all USB mics can work with phones, but here is a decent one that might work: USB Microphone with stand
In your question you mention that you are looking for the best quality. Obviously, the best answer would be to set up an actual mixer so you can have separate microphones for the piano and the singer. They are not all that expensive, and if you are serious about recording you may well decide it is worth it. To use a mixer you would want an adapter cable for the output that can plug in to the 1/8" microphone port on the phone, plus microphones and their associated cables. (To professionally mic a piano, you want the microphone to be inside. For a grand piano, I open the top then use a microphone stand to position the microphone about a foot above the sounding board near the center)
Here is one inexpensive example: Audio Mixer - It takes some setting up but this is how the studios do it.
You asked about apps. Instead of suggesting one in particular, I'd suggest you go to the play store and look around. There are a lot available, and they can help with effects and other parameters. Try some and see if you think they help. You might find one that you like.
I have tried some but still use the stock recording app that came with my tablet.
There are also some good playback apps that can do things like boost your bass or give full equalization control. Those are good for listening to any kind of music. But in the end, I don't use them.
Instead, I picked up an excellent Bluetooth speaker: OontZ Angle 3 that provides very good sound all by itself. It is well worth the price. I use it all the time.
In summary, I would suggest you start with the lapel microphone first, then go from there. 1) you know for sure it will work with your phone. 2) they are easy to use, 3) the price is right, 4) they produce surprisingly good recordings.
In the final analysis, nothing beats getting a good recording in the first place, so step one is to find a decent microphone that suits you. But playback is also important, so step two is to get a good speaker system that enhances the sound.