In GR we use the Riemannian manifold without any torsion to describe the theory. Hence, the geodesic equation can be interpreted as "a trajectory of a free falling particle" or "equation of motion". But in Einstein-Cartan manifold the torsion prevents us to use the geodesic equation as an equation of motion. So, my question is that, what is the meaning of a geodesic equation in this theory?
Torsion doesn't prevent us to use the geodesic equation as an equation of motion. When torsion is present, we need to distinguish two types of "motion": The geodesic is a curve of extremal proper time. It comes from a lagrangian formulation. The teleparallel curve is the one for which the tangent is parallely transported from one point to the next. Both are equivalent in the absence of torsion, but they're not the same when a general torsion is present. Currently, we don't know which one a massive particle should follow. The inertia principle suggest that it should be a teleparallel curve, not a geodesic curve.
Take note that, given some initial conditions, both curves are the same when torsion is completely antisymetric, which is the case when only Dirac spinors are coupled to torsion.