A mild discrepancy is some objection to a statement or theory that experts in that field can agree that they have provided no conclusive answer to that objection. The objection could even present a paradox (e.g., the Twin paradox of special relativity, or set paradoxes for mathematics, etc).
They key to accepting mild discrepancies is that there is no convincing evidence (in the opinion of most experts in that field) that the objection carries enough weight to overthrow the confidence in the statement. There is a trade-off between confidence of the statement and theory, and the weight of the convincing evidence of the objection. There is no exact rule in making a decision in favor of the objection, but there are criteria on sorting out gray areas.
These criteria include: success of the statement or theory in making successful predictions, the explanatory success of the statement or theory to account for the vast majority of the phenomena in question, a comparison of evidence in favor of the objection versus the vast amount of evidence to suggest that the objection does not present enough evidence to be of a major concern (i.e., extraordinary claims require extraordinary evidence), the mathematical beauty of the statement (if applicable) versus the mathematical beauty of the objection (if applicable), whether the objection is centered on claims of our ignorance versus claims centered on our knowledge, whether the objection hints or could possess logical fallacies (e.g., Liar's paradox), etc... etc...
Since there are so many factors that determine a mild discrepancy, it is often difficult for non-experts to appreciate the reasons why mild discrepancies are not taken more seriously. Also, dogma to particular radical/excessive ideologies can blind or hinder someone by having them inflate mild discrepancies or misclassify a major discrepancy as a mild discrepancy. Perhaps this is where many skeptics of major scientific theories are most likely to be found. They have strong enough ideologies that they are likely to discount the evidence in favor of a major scientific theory (e.g., how time is understood in general relativity, etc) and go off constructing their half-cocked pseudo theory. Often they speak in terms of the dogma of science to suggest that it is the whole scientific community that is half-cocked which is why they don't see the seriousness of their objections. They might even cite past experiences where the scientific community made corrections to their theory, or cite the healthy scientific skepticism that accompanies much of scientific dialogue (although these citations are usually made out of context and the implications are exaggerated).
Due to the sensitivity in seeing these issues clearly, it is probably not too difficult to get labelled as a crockpot if you take objections more seriously than what the community of experts feel is appropriate. It is also easy to be seen as such a diehard in a previously held theory that you are considered a dinosaur after the asteroid already hit. Science progresses despite these extreme views.
Warm regards, Harv