The Polya-Vinogradov's inequality states that if $\chi$ is a primitive character mod $q$ then $|\sum_{n\le x}\chi(n)|\le \sqrt{q}\log q$. I now want to give the estimation for the sum over primes, i.e. $\sum_{p\le x}\chi(p)$ and wonder if we can apply the Polya-Vinogradov's method?
It is proved in the book of H. Iwaniec and E. Kowalski (Corollary 5.29) that $\sum_{p\le x}\chi(p)<< \sqrt{q}x(\log x)^{-A}$ for any $A>0$ but the proof is somewhat complicated. Can we have the simpler proof for this fact?