Use the "small power" signal from the speaker output to trigger something like a 555 Timer set up as a one-shot pulse of whatever time you want to hold open a relay, either by driving the relay directly if the small output of the timer IC is enough, or by using a high side MOSFET or BJT as a load switch from the 9V you have available.
The audio style signal will not really be useful for direct amplification to activate a relay switch, unless you like musical relays!
Your relay specifications are quite important for this, so make sure you give it the required current to hold it open. You may also not even need a relay, whatever your relay is doing could probably be done easier and straight to the point using a MOSFET load switch.