We introduce a new subclass M(α, β) of close-to-convex harmonic mappings in the unit disk, which originates from the work of P. Mocanu on univalent mappings. We also give coefficient estimates, and discuss the Fekete-Szegő problem, for this class of mappings. Furthermore, we consider growth, covering and area theorems of the class. In addition, we determine a disk |z| in which the partial sum sm,n(f)(z) is close-to-convex for each function of the class M(α, β). Finally, for certain values of the parameters α and β , we solve the radii problems related to starlikeness and convexity of functions of this class.