Is there cognizance or meta-awareness of classical conditioning?
I'm not sure what you're asking.  If it's found that there's awareness of the relationships in the experiment then typically it's argued that's not conditioning and the behaviour is modified through insight.  It's been a bone of contention with respect to classical conditioning and humans in the past.  You might want to look at Lovibond and Shanks (2002).  I think some of the thinking there is wrongheaded (how you assess awareness).  Nevertheless, it's a start for looking into the topic.

If you're asking if there's an awareness of the effects of being conditioned separate from those of stimulus and response correlations, I don't think anyone has ever tested that.

Lovibond, P. F. and Shanks, D. R. (2002). The role of awareness in Pavlovian conditioning: Empirical evidence and theoretical implications. Journal of Experimental Psychology: Human Perception and Performance, 28(1):3–26.