AbstractThere 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.