Low amp/volt relay for microphone switching
It's unlikely that your mic switch opens the mic circuit as you've shown because this will leave the input unterminated, picking up stray signals and producing a hum or noise. More likely:

= Releasing the switch short-circuits the mic signal to ground.

or


The microphone is a two-way radio style unit with a push-to-talk button which is separate from the audio circuit (although it may share the screen for both).








simulate this circuit &ndash; Schematic created using CircuitLab

Figure 1. Standard audio microphone with mute switch. Figure 2. PTT (push-to-talk) mic sharing switch and audio ground.

ThreePhaseEel's comment is trying to establish which setup you have. The PTT style gives you more flexibiiity in your solution.

A hard switching of the audio can be hard on the ears due to the sudden discontinuity. The VOX (voice operated switch) unit suggested by Richard will suffer this problem too if used to switch a relay.

A more elegant solution (for the listener) would be to use an audio "ducker" as used by DJs and building paging systems (to mute background music), so called as it 'ducked' the music level below the voice level. If you can't find a ready-made device, you should be able to find a suitable circuit on a web search. You'll also find terms describing how quickly it ducked - too sudden sounds as bad as relay switching - and the release time - how long the silence has to be before the music starts to fade back up - and the release rate - how quickly it fades up once it has started.