I am in need of a (relatively) general sieve with two parameters $y, z$. I am quite sure that on the literature there must be some result of the kind that I have in mind, probably a corollary of the large sieve inequality and some estimates of average values of multiplicative function. However, I did not find it.

For each prime number $p$, let $\Omega_p \subsetneq \{0, 1, \ldots, p - 1\}$ be a set of residues modulo $p$. Denote by $\Omega$ the whole family of the $\Omega_p$'s. Suppose that $|\Omega_p| \leq c$ for all $p$, and that $$\sum_{p \leq x} |\Omega_p| \cdot \frac{\log p}{p} = k \log x + O(1),$$ for all $x > 1$, where $c, k > 0$ are given constants.

**My question is**: In which ranges (respect to $x$) should $y$ and $z$ be in order to having
$$(1) \quad |\{n \leq x : (n \bmod p) \notin \Omega_p\;\forall p \in [y, z]\}| \ll_{\Omega} \frac{x}{(\log x)^k},$$
for all $x > 1$ ? If that inequality is not possible at all, you can replace the exponent $k$ by some $k + o(1)$, as $x \to +\infty$. Anyway, I would like at least $y > (\log x)^\delta$, for fixed $\delta > 0$.

Note that, by the large sieve inequality, (1) would follow from $$\sum_{n \leq z} f_{y,z}(n) \gg_{\Omega} (\log x)^k ,$$ where $f_{y,z}$ is the multiplicative function supported on the squarefree integers with prime divisors in $[y, z]$ and such that $$f_{y,z}(p) = \frac{|\Omega_p|}{p - |\Omega_p|} .$$

Thank you in advance for any idea/reference.

**EDIT:** In light of Jan-Christoph Schlage-Puchta answer I edited the question in order to relax the hypothesis.