The distribution of the values of Dirichlet characters over the sequence of shifted primes

The talk is devoted to the following result of the speaker:
Theorem. Let $q$ be a sufficiently large natural number, and suppose that $\chi_{q}$ is a primitive character modulo $q$. Suppose also that $(l,q)=1$, and let $\varepsilon$ be arbitrary small positive constant, $\mathcal{L}\,=\,\ln q$, $x\geqslant q^{\,5/6+\varepsilon}$. Then we have:
$$ T(\chi_q )=\sum_{p\,\leqslant\, x}\chi_q(p-l)\ll x\exp\bigl(-\sqrt{\mathcal{L}}\bigr). $$

This estimate improves the result of J.B. Friedlander, K. Gong and I.E. Shparlinski (2010), which is non -trivial only for $x\geqslant q^{\,8/9+\varepsilon}$.