  We study the motion of uncharged particles and photons in the background of a
magnetically charged Euler-Heisenberg (EH) black hole (BH) with scalar hair.
The spacetime can be asymptotically (A)dS or flat. After investigating particle
motions around the BH and the behavior of the effective potential of the
particle radial motion, we determine the contribution of the BH parameters to
the geodesics. Photons follow null geodesics of an effective geometry induced
by the corrections of the EH non-linear electrodynamics. Thus, after
determining the effective geometry, we calculate the shadow of the BH. Upon
comparing the theoretically calculated BH shadow with the images of the shadows
of M87* and Sgr A* obtained by the Event Horizon Telescope collaboration, we
impose constraints on the BH parameters, namely the scalar hair ($\nu$), the
magnetic charge ($Q_{m}$) and the EH parameter ($\alpha$).
