What device are the headphones plugged into? Laptop? Tablet or Smartphone? It would help if you said at least that much or a specific model. Some computers disable their intel audio codec until it is needed so you do get a click or pop when it comes on. Realtek audio codecs do the same but without the evident noise.
It's feasible for a tablet or smartphone to do similar.
Depends if its actually clicking or if it's just turning off the driver. When the driver is on you get a very quiet hum, barely audible, but when it turns on and off, in comparison to complete silence you may interpret this rapid change in noise as a click.
Definitely codec power saving. There's similar from my Samsung A7 tablet, I have very sensitive hearing. It is a brief pop but not overly intrusive. I guess how loud the pop is to you depends what volume you have your headphones at. I have mine quite low at about 40%. There is a faint hiss in the background during quiet moments of audio playback