  Quantum transport in a non-equilibrium setting plays a fundamental role in
understanding the properties of systems ranging from quantum devices to
biological systems. Dephasing -- a key aspect of out-of-equilibrium systems --
arises from the interactions with the noisy environment and can profoundly
modify transport features. Here, we investigate the impact of dephasing on the
non-equilibrium steady-state transport properties of non-interacting fermions
on a one-dimensional lattice with long-range hopping ($\sim
\frac{1}{r^\alpha}$). We show the emergence of distinct transport regimes as
the long-range hopping parameter $\alpha$ is tuned. In the short-range limit
($\alpha \gg 1$), transport is diffusive, while for the long-range limit
($\alpha \sim \mathcal{O}(1)$), we observe a super-diffusive transport regime.
Using the numerical simulation of the Lindblad master equation, and
corroborated with the analysis of the current operator norm, we identify a
critical long-range hopping parameter, $\alpha_c \approx 1.5$, below which
super-diffusive transport becomes evident that quickly becomes independent of
the dephasing strength. Interstingly, within the super-diffusive regime, we
find a crossover from logarithmic to power-law system-size dependence in the
non-equilibrium steady-state resistance when $\alpha$ varies from $\alpha \leq
1$ to $\alpha \lesssim 1.5$. Our results, thus, elucidate the intricate balance
between dephasing and unitary dynamics, revealing novel steady-state transport
features.
