  Generalized distribution amplitudes (GDAs) of mesons can be probed by the
reactions $e^- e^+ \to M_1 M_2 \gamma$, which are accessible at
electron-positron colliders such as BESIII and Belle II. After discussing the
neutral meson production case in the first paper of this series
\cite{Pire:2023kng}, we discuss here the complementary case of the charged
meson ($M^+ M^-$) production, where one can extract the complete information on
GDAs from the interference of the amplitudes of the two competing processes
where the photon is emitted either in the initial or in the final state.
Considering the importance of the charged meson production, we present a
complete expression for the interference term of the cross section which is
experimentally accessible thanks to its charge conjugation specific property.
We adopt two types of models for leading twist $\pi \pi$ GDAs to estimate the
size of the interference term in the process $e^- e^+ \to \pi^+ \pi^- \gamma$
numerically, namely a model extracted from previous experimental results on
$\gamma^* \gamma \to \pi^0 \pi^0$ at Belle and the asymptotic form predicted by
QCD evolution equations. We include in the calculation the kinematical power
suppressed (sometimes called kinematical higher-twist) corrections up to
$1/Q^2$ for the helicity amplitudes. Both models of GDAs indicate that the
kinematical corrections are not negligible for the interference term of the
cross section measured at BESIII, thus it is necessary to include them if we
try to extract the GDAs precisely. On the other side, the kinematical
corrections are very tiny for the measurements at Belle II, and the leading
twist-2 formula of the interference term will be good enough to describe the
charge conjugation odd part of the differential cross section.
