Bounds of Multiplicative Character Sums over Shifted Primes

For an integer $q$, let $\chi $ be a primitive multiplicative character mod $q$. For integer $a$ coprime to $q$, we obtain a bound of the form $\bigl |\sum _{n\le N}\Lambda (n)\chi (n+a)\bigr |\le N/q^\delta $, $N\ge q^{3/4+\varepsilon }$, where $\Lambda (n)$ is the von Mangoldt function. This improves on a series of previous results.