Bounds for the norm of certain additive character sums

Question

Let $q$ be a power of a prime, let $\alpha$ be a primitive element in the finite field with $q$ elements and let $\chi$ be the canonical additive character of the field. Recently I became interested in bounds for the norm of sums of the type \begin{equation} \sum_{d \mid N} \mu(d) \eta_k^{(d,q)} = \sum_{\substack{k=1 \\ \gcd(k,N) =1}}^{q-1}\chi\left(\alpha^k \right), \end{equation} where $N$ is a positive divisor of $q-1$, and $\eta_k^{(d,q)}$ is the $k$-th Gaussian period of type $(d,q)$ defined by $$ \eta_k^{(d,q)} = \sum_{i=1}^{\frac{q-1}{d}} \chi\left( \alpha^{k + di} \right). $$ One could obtain bounds for these sums, say, by writing the Gaussian periods in terms of Gaussian sums $S(\psi)$, where the $\psi$ are multiplicative characters of the field, and then use the Triangle Inequality together with the fact that $|S(\psi)| = \sqrt{q}$ for non-trivial $\psi$. However I think these bounds would not be good as they do not account for all the possible changes in sign of the many terms in the sums. Another trivial bound is of course $\frac{q-1}{N}\phi(N)$, the number of terms in the right hand side above (since character images have norm $1$). Could one do better than this last bound? Are there any bounds known in literature? Thanks a lot.