  Let $\mu$ be the M\"{o}bius function and let $k \geq 1$. We prove that the
Gowers $U^k$-norm of $\mu$ restricted to progressions $\{n \leq X: n\equiv
a_q\pmod{q}\}$ is $o(1)$ on average over $q\leq X^{1/2-\sigma}$ for any $\sigma
> 0$, where $a_q\pmod{q}$ is an arbitrary residue class with $(a_q,q) = 1$.
This generalizes the Bombieri-Vinogradov inequality for $\mu$, which
corresponds to the special case $k=1$.
