  This is the second installment of a series of three papers in which we
describe a method to determine higher-point correlation functions in one-loop
open-superstring amplitudes from first principles. In this second part, we
study worldsheet functions defined on a genus-one surface built from the
coefficient functions of the Kronecker--Einsenstein series. We construct two
classes of worldsheet functions whose properties lead to several simplifying
features within our description of one-loop correlators with the pure-spinor
formalism. The first class is described by functions with prescribed
monodromies, whose characteristic shuffle-symmetry property leads to a
Lie-polynomial structure when multiplied by the local superfields from part I
of this series. The second class is given by so-called generalized elliptic
integrands (GEIs) that are constructed using the same combinatorial patterns of
the BRST pseudo-invariant superfields from part I. Both of them lead to compact
and combinatorially rich expressions for the correlators in part III. The
identities obeyed by the two classes of worldsheet functions exhibit striking
parallels with those of the superfield kinematics. We will refer to this
phenomenon as a duality between worldsheet functions and kinematics.
