The new estimate for the sum of the values of a primitive Dirichlet character modulo an integer q has been obtained over the sequence of shifted primes p − l, (l, q) = 1, p ≤ x. This estimate is nontrivial for x ≥ q 5 6 +ε and refines the estimate obtained by J. B. Friedlander, K. Gong, I. E. Shparlinskii. Their estimate holds provided that x ≥ q 8/9+ε
