  The interaction between Active Galactic Nuclei (AGN) and their host galaxies
is scarcely resolved. Narrow-line Seyfert 1 (NLS1) galaxies are believed to
represent AGN at early stages of their evolution and allow to observe AGN
feeding and feedback processes at high accretion rates. We apply a
spectroastrometric analysis to VLT MUSE NFM-AO observations of Mrk 1044, a
nearby super-Eddington accreting NLS1. This allows us to map two ionised gas
outflows traced by [O$\,$III] which have velocities of $-560\pm20\,{\rm
km\:s}^{-1}$ and $-144 \pm 5 \,{\rm km\:s}^{-1}$. Both outflows are spatially
unresolved and located close to the galaxy nucleus ($<1\,{\rm pc}$). They have
gas densities higher than $10^5\,{\rm cm}^{-3}$, which implies that the BPT
diagnostic cannot be used to constrain the underlying ionisation mechanism. We
explore whether an expanding shell model can describe the velocity structure of
Mrk 1044's unresolved multi-phase outflow. A kinematic analysis suggests that
significant turbulence may be present in the ISM around the nucleus, which may
lead to a condensation rain, potentially explaining the efficient feeding of
Mrk 1044's AGN. We identify an additional ionised gas outflowing component that
is spatially resolved. It has a velocity of $-211 \pm 22 \,{\rm km\:s}^{-1}$
and a projected size of $4.6 \pm 0.6 \,{\rm pc}$. Within the innermost 0.5"
(160$\,{\rm pc}$) around the nucleus we detect modest star formation hidden by
the beam-smeared emission from the outflow, which suggests that Mrk 1044's AGN
phase set on recently. We estimate that the multi-phase outflow has been
launched $<10^4 \,{\rm yrs}$ ago. It carries enough mass and energy to impact
the host galaxy star formation on different spatial scales, highlighting the
complexity of the AGN feeding and feedback cycle in its early stages.
