Short exponential sums over primes

I.M. Vinogradov was the first who obtained non-trivial estimates of the sum of values of non-principal character over the sequence of shifted primes. He proved that if $q$ is an odd prime number, $(l,q)=1$, $\chi$ is a non-principal character modulo $q$, then
$$ T(\chi)\,=\,\sum_{p\leqslant x}\chi(p-l)\,\ll\,x^{1+\varepsilon}\left(\sqrt{\frac{1}{q}+\frac{q}{x}}\,+\,x^{-1/6}\right). $$
This estimate is non-trivial for $x\gg q^{1+\varepsilon}$ and yields to the asymptotic formula for the number of quadratic residues (non-residues) $\pmod q$ of the form $p-l$, $p\leqslant x$. The best result here belongs to A.A. Karatsuba. In 1970 he proved that if $q$ is a prime number, $\chi$ is a non-principal character modulo $q$ and $x\geqslant q^{\,1/2+\varepsilon}$, then the following estimate holds:
$$ T_{1}(\chi)\,\ll\,xq^{-\,\varepsilon^{2}/1024}. $$
In 2013, the author obtained non-trivial estimate of $T(\chi_{q})$ for composite $q$ and primitive character in the case $q^{\,5/6+\varepsilon}$. In the talk, we present the following new theorem:
Theorem. Let $D$ be a sufficiently large natural number, $\chi$ is a non-principal character modulo $D$, $\chi_q$ is a primitive character modulo $q$ that induces character $\chi$, $q$ is a cubefree number, $(l ,D)=1$. Then the following estimate holds
$$ T(\chi)\,=\,\sum_{n\leqslant x}\Lambda(n)\chi(n-l)\,\ll\,x\exp{\bigl(-0.6\sqrt{\ln{D}}\bigr)}, $$
for $x\ge D^{\,1/2+\varepsilon}$ (the constant in $\ll$ depends only on $\varepsilon$).