Prime and Möbius correlations for very short intervals in $F_p[x]$

We investigate function field analogs of the distribution of primes,
and prime $k$-tuples, in "very short intervals'' of the form $I(f) :=
\{ f(x) + a : a \in F_p \}$ for $f(x) \in F_p[x]$ and $p$ prime, as
well as cancellation in sums of function field analogs of the Möbius
$\mu$ function and its correlations (similar to sums appearing in
Chowla's conjecture).

For generic $f$, i.e., for $f$ a "Morse polynomial", we show that
error terms are roughly of size $O(\sqrt{p})$ (with typical main terms
of order $p$).  We also give examples of $f$ for which there is no
cancellation at all, and intervals where the heuristic ``primes are
independent'' fails very badly.

Time permitting we will discuss the curious fact that (square root)
cancellation in Möbius sums is equivalent to (square root)
cancellation in Chowla type sums.