Sum over dyadic interval with von Mangoldt function to sum over primes

Question

Let $1<c<2$ and write $\gamma=1/c$ so that $1/2<\gamma<1$. Now write $\lfloor u\rfloor=u-\psi(u)-1/2$ so that $\psi(u)=\{u\}-1/2$, where $\{u\}$ is the fractional part of $u$. We then denote by $\Lambda(n)$, the von Mangoldt function $$\Lambda(n)=\begin{cases} \log p & \text{ if } n=p^r \text{ for some prime } p \text{ and some integer } r\ge 1, \\ 0 & \text{ otherwise.}\end{cases}$$ I want to show that if $$\sum_{x/2<n\le x}\Lambda(n)(\psi(-(n+1)^\gamma)-\psi(-n^\gamma))\ll x^{\gamma-\kappa}$$ for some $\kappa>0$ then $$\sum_{p\le x^c}(\psi(-(p+1)^\gamma)-\psi(-p^\gamma))\ll x^{1-\kappa'},$$ for some $\kappa'>0$; the sum is over primes $p\le x^c$. I presume this is an exercise in partial summation but I am unsure of how to proceed.