Based on results on Hurwitz-Brill-Noether theory obtained by H. Larson we give a picture of the irreducible components of Wr d (C) for a general k-gonal curve of genus g. This picture starts from irreducible components of Wr d (C) restricted to an open subset of Pic(C) satisfying Brill-Noether theory as in the case of a general curve of genus g. We obtain some degeneracy loci associated to a morphism of locally-free sheaves on them of the expected dimension. All the irreducible components of the schemes Wr d (C) are translates of their closures in Pic(C). We complete the proof that the schemes Wr d (C) are generically smooth in case C is a general k-gonal curve (claimed but not completely proved before). We obtain some results on the tangent spaces to the splitting degeneracy loci for an arbitrary k-gonal curve and we obtain some new smoothness results in case C is a general k-gonal curve.
