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.
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