I’m not happy that I am rolling dice here. I mean, it’s advertised for triggering sound effects, so I would assume that anyone testing the design would have noticed the click and how disruptive it is. I’ll probably try it, but this is kind of a big design flaw.
I also have observed the click problem. It definitely is audible only on the speaker output, not the earphone jack.
I did a bit of surgery on the board, cutting the trace to pin 1 (SHUTDOWN) of the amplifier, on the top of the board between the pin and the board via. Then I connected pin 1 to +3.3 V to keep the amplifier enabled, because it is disabled with pin 1 floating. It appears doing it this way will keep the pullup resistor in the circuit to allow the PLAYING signal to otherwise function normally. Cutting the trace is difficult; connecting pin 1 to + 3.3 V is even harder.
The good news is that this eliminates the click. The bad news is that with the amplifier always enabled, the speaker emits a distinct low-level whine when no MP3 is playing. When you are playing an MP3, even if it’s just a recording of a bit of noise, you can’t hear the whine. Take your pick, but I think I prefer the click.