It sounds like you could have something going on in your throat or voice box. An ENT specialist should examine these structures. If this specialist can hear the clicking, that will help to locate where it is coming from.
Your description of a "clicking or popping" feeling caught my attention. I experience this as a sound (as well as a sensation) when breathing in. But the sound is not very loud and seems to be related to body posture. Also, I often hear a "sputter" or snore-like noise when I breath in deeply. I am diagnosed with idiopathic diaphragmatic paralysis (right side). These sounds and sensations are related to my condition but I don't know how they are produced. For many years prior to diaphragmatic paralysis I was known to have diaphragmatic "eventration", a slight elevation on one side of the diaphragm. I know from experience that doctors will often "pass over" this condition because "eventration" is known to be common and normally produces no symptoms. You may want to ask your doctor to take another look at your recent x-rays. To my experience, the sensations you describe do not represent a serious threat. But even so, there should be an explanation for "clicking or popping". I hope this information will be of some help. Please let us know how this turns out.