I wish to hook up the line-out from a digital piano to a surplus 4 ohm 15w speaker unit. I bought a simple Amplifier module and connected it up.
My plan was to roughly assemble the circuit and, if it worked ok, transfer to strip-board with RCA connector for Audio and a DC power socket and some sort of quick connector for speaker cable.
When I play a note on the piano I get a good level of sound from the speaker but the quality is very poor.
I guess I need to use better wiring on the input side of the amp?
Circuit:
 
Here's a photo:
 
Battery: 12V SLA 3.3Ah
Speaker: 4 Ohm 15W
Amplifier: 
 TI amp TAS2552EVM 4.5-12V, 3.5W "Music power", for 4-15 ohm speaker, 40Hz-20kHz,
Source: Line-out 1/4" TR connector (left channel, set to mono).
Pot: 10K Linear
The speaker is a rear satellite speaker unit from an old TV that had 5.1 surround-sound. The speaker enclosure contains two sounders (approx 5" and 1").
The digital piano is powered through a mains adapter (100-240 VAC 0.2A 50-60Hz) connected using a USB cable (5V 1A). It is light so assume switched mode.
Notes sound rather buzzy. There is mains hum in the background.
What steps should I do to improve the quality of the sound?
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You are the best.
Get a real amplifier.  A car stereo produces 17.5 watts per channel.  Your amplifier produces 3.5 watts.
Get a Car stereo with an aux input jack.  Run the piano into the aux jack.
      An amplifier with Automatic level control may help you out.