Q1 is pulsing, but its collector is held at a constant low by C1. If you connect the Q4 gate directly to the Q1 emitter, Q4 will turn on the relay when the signal is present. The LED can be connected in parallel with the relay output, eliminating Q2 and Q3. Add a protection diode across the relay output to GND.
R2-C1 has a corner freq of 34 Hz, plenty low enough to filter a 20 kHz signal. The 4.7 ms time constant sets the delay between when the signal disappears and when the relay releases.
ak