You built a multi-vibrator which is designed to oscillate. What did you expect? You can't modify it to do what you want.
You need to replace the multi-vibrator with a 555 connected as a one-shot and a Flip-Flop such as a 4013.
Connect the 555 as a one-shot with a period of about 0.5s. (Google "555 timer" if you need a circuit).
Connect the 555 trigger input to the Mic amp output (with a 10k pull-up resistor to +9V).
Connect the 555 output to the FF Clk input.
Connect the FF D input to the FF /Q output.
Connect all unused FF inputs to ground.
Connect the Q output to the relay driver transistor.
The 555 one-shot provides a delay so the FF will not be triggered more than once for each clap. Each clap such then trigger the FF into the alternate state (Off to On to Off to...).