A thought:
Can you reliably reproduce the sound?
In the past, I had a click on the A. It happened when I was playing the D and the A was still being held down by my 3rd (ring) finger in a barre chord-like position, though I was only barring 2 strings.
I had set out to remove ALL extraneous noise from my playing, so this was really bugging me.
I did all sorts of set-up tweaks to get rid of it. I could pretty much get rid of by getting the action high enough, but that was not a very happy solution.
However, once I was able to reliably reproduce it, I traced it to the A string striking the fret just above where my 3rd finger was depressing the A, even though I was actually playing a pitch on the D string. After plucking the D, my finger came back against the A. Normal technique, yes, and not a problem, but while barring both the D and A, the A was clicking on the fret because the A was not fully depressed, but was in a sort of "muted" state. The muting worked to keep the string from ringing, but not from clicking on the fret.
I changed my technique just slightly to make sure the A string stayed quiet. It is hard to describe now just what I changed, I've forgotten. I'll try to figure it out and post it, if I can.
My point is to look and listen very carefully. Isolate the exact circumstances in both hands when it happens and where the sound is exactly coming from. Get so you can do it "on demand." If you can do that, you can probably determine if it is a techique issue or not.
