Noise on piezo analog read due to Arduino power supply

The 9V wall adapter is in most cases a switched power regulator with a high frequency. That's like a radio transmitter direct beside the wire. So there is nothing mystical, when the smooth USB Voltage does not show problems but the "radio transmitter" does.

There are two ways the power supply can impact: over radio wave or as ripples over the regulated Voltage of your Arduino.

I asked myself, which one it is (or even both)? If you use the extra regulated analog voltage reference (AREF) as the +5V on your Voltage divider, it would be more likely that the problem comes through the air. Then it would help if you activate the shield of the wire by connecting it to GND.

If a rippled supply voltage is the problem, you have to smooth it. Perhaps with a low pass filter (a big Elko between Vcc and GND).

Perhaps you have a solenoid iron core. Then you can wrap the wire to the piezo sensor a few times around it (near the A0 Pin). This technique is often used in computer power supplies to prevent the switching frequencies from escaping to the public power lines.