  We show that the expected asymptotic for the sums $\sum_{X < n \leq 2X}
\Lambda(n) \Lambda(n+h)$, $\sum_{X < n \leq 2X} d_k(n) d_l(n+h)$, and $\sum_{X
< n \leq 2X} \Lambda(n) d_k(n+h)$ hold for almost all $h \in [-H,H]$, provided
that $X^{8/33+\varepsilon} \leq H \leq X^{1-\varepsilon}$, with an error term
saving on average an arbitrary power of the logarithm over the trivial bound.
Previous work of Mikawa, Perelli-Pintz and Baier-Browning-Marasingha-Zhao
covered the range $H \geq X^{1/3+\varepsilon}$. We also obtain an analogous
result for $\sum_n \Lambda(n) \Lambda(N-n)$.
  Our proof uses the circle method and some oscillatory integral estimates
(following a paper of Zhan) to reduce matters to establishing some mean-value
estimates for certain Dirichlet polynomials associated to "Type $d_3$" and
"Type $d_4$" sums (as well as some other sums that are easier to treat). After
applying H\"older's inequality to the Type $d_3$ sum, one is left with two
expressions, one of which we can control using a short interval mean value
theorem of Jutila, and the other we can control using exponential sum estimates
of Robert and Sargos. The Type $d_4$ sum is treated similarly using the
classical $L^2$ mean value theorem and the classical van der Corput exponential
sum estimates.
