Well, for the mp3 sound module ... does it turn on and default to the audio track so that you only have to press play? Could you maybe configure to play automatically on power up?
There are modules which you have to "talk" to them through UART sending specific commands.
If the module has buttons, then the buttons work by connecting two wires on the circuit board together, one wire is either going to ground pulling down the input/output wire to ground, or the wire is connected to some voltage so that when the button is pressed, that voltage goes to the input/output pin of the chip - that's how the chip detects a button is pressed.
So if there's some buttons, you could connect your wire to that button and simulate the button being pressed using your microcontroller.
Often people use cheap mp3 players precisely for this reason, instead of mp3 modules ... well, besides price.
For example see this at 3.66$ each :
https://www.ebay.com/itm/144092272329Has a on-off slide switch, so you can permanently set it to on and just disconnect battery using a npn transistor or mosfet to turn it off or on completely.
It will default to first mp3 file from the sd card or internal usb memory, because there's no screen or anything, so you could just have your controller turn on power to the player a few seconds before you want sound to play, then press the play button as i explained in the above paragraph... when you want to stop you can just disconnect power.