  We consider the asymptotic behavior of small global-in-time solutions to a 1D
Klein-Gordon equation with a spatially localized, variable coefficient
quadratic nonlinearity and a non-generic linear potential. The purpose of this
work is to continue the investigation of the occurrence of a novel modified
scattering behavior of the solutions that involves a logarithmic slow-down of
the decay rate along certain rays. This phenomenon is ultimately caused by the
threshold resonance of the linear Klein-Gordon operator. It was previously
uncovered for the special case of the zero potential in [50]. The Klein-Gordon
model considered in this paper is motivated by the asymptotic stability problem
for kink solutions arising in classical scalar field theories on the real line.
