Racial or ethnic bias on the part of your polygraph operator could certainly influence the outcome of your polygraph examination.
With respect to breathing through your mouth rather than your nose, I don't recall any doctrine on that in the polygraph literature.
Certainly, one can breathe through both the nose and mouth simultaneously, particularly if you normally do so. I would also be mindful, however, of negative stereotypes about "mouthbreathers."
I wish you all the best if you go through with the polygraph. Note that our best advice, as reflected in The Lie Behind the Lie Detector, is to not submit in the first place, but rather to seek employment some place else where your honesty and integrity won't be judged based on such pseudoscientific nonsense as polygraphy.