  In this paper we investigate effects of the $r$-mode instability on a newborn
rapidly-rotating magnetar with fall-back accretion. Such a magnetar could
usually occur in core-collapse supernovae and gamma-ray bursts. We find that
the magnetar's spin and $r$-mode evolution are influenced by accretion. If the
magnetar is sufficiently spun up to a few milliseconds, gravitational radiation
leads to the growth of the $r$-mode amplitude significantly. The maximum
$r$-mode amplitude reaches an order $\sim 0.001$ when the damping due to the
growth of a toroidal magnetic field balances the growth of the $r$-mode
amplitude. If such a sufficiently spun-up magnetar was located at a distance
less than 1\,Mpc, then gravitational waves would be detectable by the Einstein
Telescope but would have an extremely low event rate. However, if the spin-up
is insufficient, the growth of the $r$-mode amplitude is mainly due to the
accretion torque. In this case, the maximum $r$-mode amplitude is of the order
of $\sim 10^{-6}-10^{-5}$.
