UPDATE 06-Nov-2019! I have found a way to remove the click in firmware. This is released in QCX firmware version T1.02. See below for details. 

Hardware mod

This modification by Karlis YL3JG improves the sound of the sidetone at low volume. The QCX microcontroller injects a 700Hz tone to the audio signal path. The pulse width (duty cycle) is used to set the volume. So for the highest volume (setting 99 in the Sidetone Volume menu), the duty cycle is 50%. It is logarithmically reduced as the gain is reduced. Many people find that the full volume sidetone is too loud and use a much lower setting.

Karlis noted that the start and end of the sidetone carries a click, which is worse at lower sidetone volumes. The reason for this is that the wiper of R27 has a 2.5V bias. During receive, the microcontroller pin is at high impedance. When the sidetone is enabled, the microcontroller pin outputs 700Hz with a duty cycle dependent on volume setting; this has an average which is effectively setting a new DC bias level at the R27 wiper. This sudden change in DC bias causes the audible "click". At full volume, the duty cycle is near 50%, resulting in a DC bias near 2.5V anyway; therefore there is little or no change in DC level when the sidetone starts, and no click. At low volume the change in DC bias is substantial, and the low sidetone volume also means that the click is proportionately louder. 

Karlis inserted a 0.1uF (100nF, "104") capacitor in series with R59. This provides DC isolation and eliminates the DC spike that causes the "click". The exact value of this capacitor is not critical. It is easy to insert this capacitor in series without any PCB modification; simply cut the long lead of resistor R59 and connect the capacitor across the gap. 

The following oscilloscope screenshots show a 24-wpm DIT in semi-QSK mode. The oscilloscope probe was connected at the QCX audio output and the audio gain at about 60%. The sidetone volume configuration menu setting is 99, 75, 50 and 25 respectively. In all cases the "before" modification image is on the left, and the "after" modification image is on the right. CLICK the images for full-size versions. You can see clearly that at 50 and 25 volumes, there is a huge spike at the beginning and end of the sidetone. With the capacitor in place, the spike is eliminated. It makes very little difference at high volume, but a huge difference at low volume.

Note also that the effective volume is slightly changed by this modification; at 75 the volume appeared to be increased by the modification but at 50 and 25, somewhat reduced; this is not a problem, it just slightly changes a graph of perceived volume vs volume setting, which was in any case not perfectly logarithmic. 

Firmware modification

In version T1.02 of the QCX firmware I have managed to remove this click entirely by firmware changes, no hardware modification is required. Assuming a 700Hz sidetone, this solution runs a Pulse Width Modulation at a frequency of 42kHz. The duty cycle is changed from slightly above 50%, to slightly under 50%, and back again; this occurs 700 times per second (1400 duty cycle changes per second), in order to simulate a squarewave at 700Hz. 

The result is a "DC" component at 2.5V plus a 700Hz audio signal superimposed. Harmonics and the 35kHz signal are filtered out by the CW filter. Since the DC component is the same as the existing 2.5V DC bias signal in the receiver signal chain, on starting/stopping the sidetone there are no DC discontinuities and hence NO CLICKS associated with the sidetone. 

The following comparisons have the BEFORE (unmodified QCX) on the left, and v1.02 of the firmware on the right (NO hardware modifications). In these oscilloscope screenshots, a dummy load was used (no antenna connected), the QCX is in Practice Mode, and the QCX audio gain control is at maximum. You can see clearly that the spike at the start and end of the signal is eliminated, removing the click. NOTE: the signal amplitude to volume mapping is altered but this is not considered an important issue. 

Finally this is a really low volume sidetone, the volume setting is only 5. There are no clicks at all. Very clean sound!