Character Sums over Shifted Primes

We obtain a new bound for sums of a multiplicative character modulo an integer $q$ at shifted primes $p+a$ over primes $p\le N$. Our bound is nontrivial starting with $N\ge q^{8/9+\varepsilon}$ for any $\varepsilon>0$. This extends the range of the bound of Z. Kh. Rakhmonov that is nontrivial for $N\ge q^{1+\varepsilon}$.