On the Exceptional Set of the Sum of a Prime Number and a Fixed Degree of a Prime Number

Let X be a sufficiently great real number and M denote the set of natural numbers not exceeding X which cannot be written as a sum of a prime and a fixed degree of a prime number from the arithmetical progression with difference d. Let E-d(X) = cardM. We obtain a new numerical degree estimate for the set E-d(X) and an estimate from below for the number of presentations of n is not an element of M in the specified type. The proven estimates refine the generalization for an arithmetical progression of results earlier got by V.A. Plaksin.