First, let me mention that one must be careful when asserting that an implication fails. Taken literally, the assertion  "D(x) does not imply P(x)" is logically equivalent to the assertion that D(x) is true and P(x) is false. This meaning of material implication that is used in mathematics is not the same as the natural language interpretation of if-then. For example, if the professor says to a student "It is not true that if you pass the final, then you pass the class", most people would not want the students to deduce logically that he or she will pass the final, but fail the class. But this does follow logically from the mathematical usage of material implication. So your definition of "defect" may not be what you intend. A related issue is involved when there are quantifiers. When you say D does not imply P, perhaps you just mean that there is at least one x such that D(x) holds but P(x) fails. 

In any case, the unique minimal defect is simply the assertion D(x) = "either P(x) holds, or Q(x) fails". This is a defect because it does not imply P(x), except for those x for which Q(x) already implies P(x), and because if D(x) and Q(x), then P(x) follows. It is minimal, since if D'(x) is any other defect, then D'(x) ∧ Q(x) implies P(x), and so D'(x) implies either that P(x) holds or Q(x) fails, which is exactly D(x).