Turn the latency up on the Blue Icicle. Your buffer is too low. Also possible is your wireless radio on the computer. Disable it from the Device Manager. Another hint is to make sure your Audacity session has visible a timeline long enough to record everything you want + about half again more. So for a 5 minute song make sure there is 7.5 minutes of time showing on the screen expanded out. This is because when the screen flips to follow the waveform it can cause clicks too.

Thanks. The Icicle has just one knob, for gain. Would turning up the gain accomplish the same thing? I'll try the sizing but there were clicks 'between' the screen flips too. I appreciate the advice.

Turning up the gain won't help. That isn't causing the clicks. You change the latency in the control panel.