  We present the first results from a detailed spectral-timing analysis of a
long ($\sim$130 ks) XMM-Newton observation and quasi-simultaneous NuSTAR and
Swift observations of the highly-accreting narrow-line Seyfert 1 galaxy Mrk
1044. The broadband (0.3$-$50 keV) spectrum reveals the presence of a strong
soft X-ray excess emission below $\sim$1.5 keV, iron K$_{\alpha}$ emission
complex at $\sim$6$-$7 keV and a `Compton hump' at $\sim$15$-$30 keV. We find
that the relativistic reflection from a high-density accretion disc with a
broken power-law emissivity profile can simultaneously explain the soft X-ray
excess, highly ionized broad iron line and the Compton hump. At low frequencies
($[2-6]\times10^{-5}$ Hz), the power-law continuum dominated 1.5$-$5 keV band
lags behind the reflection dominated 0.3$-$1 keV band, which is explained with
a combination of propagation fluctuation and Comptonization processes, while at
higher frequencies ($[1-2]\times10^{-4}$ Hz), we detect a soft lag which is
interpreted as a signature of X-ray reverberation from the accretion disc. The
fractional root-mean-squared (rms) variability of the source decreases with
energy and is well described by two variable components: a less variable
relativistic disc reflection and a more variable direct coronal emission. Our
combined spectral-timing analyses suggest that the observed broadband X-ray
variability of Mrk~1044 is mainly driven by variations in the location or
geometry of the optically thin, hot corona.
