There has been recent interest in a hybrid form of the celebrated conjectures of Hardy–Littlewood and of Chowla. We prove that for any
$k,\ell \ge 1$ and distinct integers
$h_2,\ldots ,h_k,a_1,\ldots ,a_\ell $, we have:
$$ \begin{align*}\sum_{n\leq X}\mu(n+h_1)\cdots \mu(n+h_k)\Lambda(n+a_1)\cdots\Lambda(n+a_{\ell})=o(X)\end{align*} $$
for all except
$o(H)$ values of
$h_1\leq H$, so long as
$H\geq (\log X)^{\ell +\varepsilon }$. This improves on the range
$H\ge (\log X)^{\psi (X)}$,
$\psi (X)\to \infty $, obtained in previous work of the first author. Our results also generalise from the Möbius function
$\mu $ to arbitrary (non-pretentious) multiplicative functions.