  There has been recent interest in a hybrid form of the celebrated conjectures
of Hardy-Littlewood and of Chowla. We prove that for any $k,\ell\ge1$ and
distinct integers $h_2,\ldots,h_k,a_1,\ldots,a_\ell$, we have $$\sum_{n\leq
X}\mu(n+h_1)\cdots \mu(n+h_k)\Lambda(n+a_1)\cdots\Lambda(n+a_{\ell})=o(X)$$ for
all except $o(H)$ values of $h_1\leq H$, so long as $H\geq (\log
X)^{\ell+\epsilon}$. 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 generalize from the M\"obius function $\mu$ to arbitrary (non-pretentious)
multiplicative functions.
