Brill-Noether loci of pencils with prescribed ramification on moduli of curves and on severi varieties on $K3$ surfaces

Under the assumption that the adjusted Brill-Noether number (Formula presented) is at least -g, we prove that the Brill-Noether loci in Mg,n of pointed curves carrying pencils with prescribed ramification at the marked points have a component of the expected codimension with pointed curves having Brill-Noether varieties of pencils of the minimal dimension. As an application, the map from the Hurwitz scheme to Mg is dominant if (Formula presented) and generically finite otherwise, settling a variation of a classical problem of Zariski. In the second part of the paper, we study the analogous loci of curves in Severi varieties on K3 surfaces, proving existence of curves with nongeneral behaviour from the point of view of Brill-Noether theory. This extends previous results of Ciliberto and the first-named author to the ramified case. We apply these results to study correspondences and cycles on K3 surfaces in relation to Beauville-Voisin points and constant cycle curves.