  This article is the second in a series of three articles, the aim of which is
to study various correspondences between four enumerative theories associated
to a surface $S$: Gromov-Witten theory of $S^{[n]}$, orbifold Gromov-Witten
theory of $[S^{(n)}]$, relative Gromov-Witten theory of $S\times C$ for a nodal
curve $C$ and relative Donaldson-Thomas theory of $S\times C$. In this article,
we focus on quasimaps to moduli spaces of sheaves on a K3 surface $S$. We
construct a surjective cosection of the obstruction theory of moduli spaces of
$\epsilon$-stable quasimaps. We then establish reduced wall-crossing formulas,
which relate reduced Gromov-Witten theory of moduli spaces of sheaves on $S$
and reduced Donaldson-Thomas theory of $S\times C$. As applications, we prove
the wall-crossing part of Igusa cusp form conjecture; higher-rank/rank-one
relative Donaldson-Thomas correspondence on $S\times C$, if $g(C)\leq1$;
relative Donaldson-Thomas/Pandharipande-Thomas correspondence on $S\times
\mathbb{P}^1$.
