Let $a,q,N$ be integers such that $N/2 \leq q \leq N$ and $a/q \notin \mathbb{Z}$.

Is the following estimate true, and, if so, how can it be proved? \[\left|\sum_{1 \leq p \leq N} \exp(2\pi i p a/q) \right|\leq |a|^{o(1)} N^{o(1)},\] where $f(x)=x^{o(1)}$ means $\lim_{x \rightarrow \infty}\frac{\log f(x)}{\log x}=0$, or equivalently $f(x) = O_{\epsilon} (x^{\epsilon})$ for all $\epsilon > 0$. Can it be obtained, for example, from Vinogradov-type estimates?

It would even be useful to know whether the estimate holds in the following average sense: $$\sum_{\substack{N/2 \leq q \leq N \\ a/q \notin \mathbb{Z}}} \left|\sum_{1 \leq p \leq N} \exp(2\pi i p a/q) \right|\leq |a|^{o(1)} N^{1+o(1)}.$$

$N$ may be assumed to be as large as required.