  We consider a one-dimensional fermionic lattice system with long-ranged
power-law decaying hopping with exponent $\alpha$. The system is further
subjected to dephasing noise in the bulk. We investigate two variants of the
problem: (i) an open quantum system where the setup is further subjected to
boundary reservoirs enabling the scenario of a non-equilibrium steady state
charge transport, and (ii) time dynamics of an initially localized single
particle excitation in the absence of boundary reservoirs. In both variants,
anomalous super-diffusive behavior is observed for $1< \alpha< 1.5$, and for
$\alpha> 1.5$ the setup is effectively short-ranged and exhibits conventional
diffusive transport. Our findings are supported by analytical calculations
based on the multiple-scale analysis technique that leads to the emergence of a
fractional diffusion equation for the density profile. Our study unravels an
interesting interplay between long-range interaction and dephasing mechanism
that could result in the emergence of unconventional behaviour in open quantum
systems.
