Circuit Idea about 555 Timer
You need to apply basic logic to the two signals you'd like to combine.  In this case, if the 555 timer output is \$A\$, and the 20 Hz clock signal is \$B\$, the specific logic you need to apply is \$A\bar{B}\$, read A and not B.

You can implement this with standard logic chips, as shown below.  The output of U1, and AND gate,  is logic HIGH when both of its inputs are logic HIGH.  U2 is a NOT gate, which is an inverter.



The logic gates can be implemented from the standard logic family, like the 7400 series.

I've included two different timing diagrams corresponding to two different 20 Hz signals.