MOSFET gate grouding via LED - noise
The only "gate-grounding mechanism" in your circuit is the MCU IO pin and, if you are hearing a weak noise thru the speaker (when the MCU output is low), then your MCU is not adequately connected to ground or your MOSFET is incorrectly wired or broken. I would also recommend that you put (maybe) 100 ohms in series with your speaker to limit the current drawn from the power rail when the MOSFET does turn on.

If, on the other hand you are floating your MCU output or have the MCU in reset, then leakage currents from the IO pin could begin to turn on the MOSFET and the sound you hear is power supply noise. The LED, D1 could allow a voltage of maybe 1.5 volts to appear on your gate under these circumstances and that is starting to be enough to turn a "standard" MOSFET on.