  We examine correlations of the M\"obius function over $\mathbb{F}_q[t]$ with
linear or quadratic phases, that is, averages of the form \begin{equation}
\label{eq:average} \frac{1}{q^n}\sum_{\text{deg }f<n} \mu(f)\chi(Q(f))
\end{equation} for an additive character $\chi$ over $\mathbb{F}_q$ and a
polynomial $Q\in\mathbb{F}_q[x_0,\ldots,x_{n-1}]$ of degree at most 2 in the
coefficients $x_0,\ldots, x_{n-1}$ of $f=\sum_{i< n}x_i t^i$. Like in the
integers, it is reasonable to expect that, due to the random-like behaviour of
$\mu$, such sums should exhibit considerable cancellation. In this paper we
show that the above correlation is bounded by $O_\epsilon \left(
q^{(-\frac{1}{4}+\epsilon)n} \right)$ for any $\epsilon >0$ if $Q$ is linear
and $O \left( q^{-n^c} \right)$ for some absolute constant $c>0$ if $Q$ is
quadratic. The latter bound may be reduced to $O(q^{-c'n}$) for some $c'>0$
when $Q(f)$ is a linear form in the coefficients of $f^2$, that is, a Hankel
quadratic form, whereas for general quadratic forms, it relies on a bilinear
version of the additive-combinatorial Bogolyubov theorem.
