Also, just to be on the safe side, let me state the sort-of-obvious: a standard tremolo effect does not modulate the signal on/off/on/off like a switch, it ramps up and down like a volume pedal, even when the effect is extreme like a hard stuttering tremolo.
Clicks in the audio path coupled from another section are often the result of current pulses in the power supply that are large enough/sudden enough to disturb the supply to the audio path. Additional decoupling in the power supply between the two parts of the circuit (control part and audio part) might help.
The simplest form of decoupling is a resistor in series with the supply voltage, followed by a capacitor to ground (RC filter.) Schematically, it'd look like this:
shared supply-> -----+-----vvvvv-----+------> Audio circuit
| about 1K |
| = about 100uF
| |
| V <- audio ground, see note
|
+-----vvvvv-----+------> control circuit
about 1K |
= about 100uF
|
V <- control ground, see note
Note: the node marked "audio ground" should meet the node marked "control ground" at one point only.
This split point should be far enough back towards the raw supply that all current loops that include
audio subcircuits should be closed on their separated side of the split, and all current loops that include control
subcircuits should be closed on their own separate side of the split.
Man, it's been a while since I made an ASCII schematic!
Oh, and good luck with maintaining your parental-omniscience credibility! From my experience, that's a hard task once they hit about 12, but your strategy of claiming specific, limited areas of expertise is (also in my experience) about the best you can hope for. Hopefully the missing "omni-" part of your omniscience will go unnoticed.