  Field perturbations of a curved background spacetime generally propagate not
only at the speed of light but also at all smaller velocities. This so-called
$Hadamard\,tail$ contribution to wave propagation is relevant in various
settings, from classical self-force calculations to communication between
quantum particle detectors. One method for calculating this tail contribution
is by integrating the homogeneous wave equation using Characteristic Initial
Data on the light cone. However, to the best of our knowledge, this method has
never been implemented before except in flat or conformally-flat spacetimes,
where null geodesics emanating from a point do not cross. In this work, we
implement this method on the black hole toy model Pleba\'nski-Hacyan spacetime,
$\mathbb{M}_2\times\mathbb{S}^2$. We obtain new results in this spacetime by
calculating the Hadamard tail of a scalar field everywhere where it is defined
(namely, in the maximal normal neighbourhood of an arbitrary point) and
investigate how it varies for various values of the coupling constant. This
serves as a proof-of-concept for the Characteristic Initial Data method on
spacetimes where null geodesics emanating from a point $do$ cross.
