I agree with @leftaroundabout, this sounds just like a problem I saw at my old middle school many years ago. Almost certainly there is a noise gate somewhere in the signal chain that is cutting out when the input signal is too low, or not responding fast enough when the signal is present.
Look for something with a "threshold" parameter. If the system is installed in a rack, it could even ben on the back of the mixer next to the input, and thus out of sight. Look for the word "threshold" on a knob, or "thresh" for short. If you lower the threshold to its minimum, or remove the gate from the chain entirely, you should see the problem go away.