Let be a K3 surface with primitive curve class . We solve the relative Gromov–Witten theory of in classes and . The generating series are quasi-Jacobi forms and equal to a corresponding series of genus Gromov–Witten invariants on the Hilbert scheme of points of . This proves a special case of a conjecture of Pandharipande and the author. The new geometric input of the paper is a genus bound for hyperelliptic curves on K3 surfaces proven by Ciliberto and Knutsen. By exploiting various formal properties we find that a key generating series is determined by the very first few coefficients. Let be an elliptic curve. As collorary of our computations, we prove that Gromov–Witten invariants of in classes and are coefficients of the reciprocal of the Igusa cusp form. We also calculate several linear Hodge integrals on the moduli space of stable maps to a K3 surface and the Gromov–Witten invariants of an abelian threefold in classes of type .