  The very latest observation of $M87$ supermassive black hole (BH) by the
Event Horizon Telescope (EHT) provides the accretion onto BHs is an interesting
study in the theory of gravity. We study the geodesics structure and accretion
near a nonlinear electrodynamics BH in strong and weak field approximations.
These approximations provide the disc-like structure under the geodesic motion
and accretion around the BH. Near the equatorial plane, we provide some new
reasons to make circular orbits and accretion of test particles around the BH.
Then we investigate perturbations, the critical speed of the fluid and the mass
accretion rate of particles around the central object. The physical validity of
this study shows that the parameter $\beta$ and $Q$ play an important role in
the circular orbits and the mass accretion rate in strong and weak field
approximations.
