  We develop an approach to study character sums, weighted by a multiplicative
function $f:\mathbb{F}_q[t]\to S^1$, of the form \begin{equation} \sum_{G\in
\mathcal{M}_N}f(G)\chi(G)\xi(G), \end{equation} where $\chi$ is a Dirichlet
character and $\xi$ is a short interval character over $\mathbb{F}_q[t].$ We
then deduce versions of the Matom\"aki-Radziwill theorem and Tao's two-point
logarithmic Elliott conjecture over function fields $\mathbb{F}_q[t]$, where
$q$ is fixed. The former of these improves on work of Gorodetsky, and the
latter extends the work of Sawin-Shusterman on correlations of the M\"{o}bius
function for various values of $q$.
  Compared with the integer setting, we encounter a different phenomenon,
specifically a low characteristic issue in the case that $q$ is a power of $2$.
  As an application of our results, we give a short proof of the function field
version of a conjecture of K\'atai on classifying multiplicative functions with
small increments, with the classification obtained and the proof being
different from the integer case.
  In a companion paper, we use these results to characterize the limiting
behavior of partial sums of multiplicative functions in function fields and in
particular to solve a "corrected" form of the Erd\H{o}s discrepancy problem
over $\mathbb{F}_q[t]$.
