  In this paper, we study the $\phi^4$ kink scattering from a spatially
localized $\mathcal{PT}$-symmetric defect and the effect of the kink's internal
mode (IM) is discussed. It is demonstrated that if a kink hits the defect from
the gain side, a noticeable IM is excited, while for the kink coming from the
opposite direction the mode excitation is much weaker. This asymmetry is a
principal finding of the present work. Similar to the case of the sine-Gordon
kink studied earlier, it is found that the $\phi^4$ kink approaching the defect
from the gain side always passes through the defect, while in the opposite case
it must have sufficiently large initial velocity, otherwise it is trapped by
the loss region. It is found that for the kink with IM the critical velocity is
smaller, meaning that the kink bearing IM can pass more easily through the loss
region. This feature, namely the "increased transparency" of the defect as
regards the motion of the kink in the presence of IM is the second key finding
of the present work. A two degree of freedom collective variable model offered
recently by one of the co-authors is shown to be capable of reproducing both
principal findings of the present work. A simpler, analytically tractable
single degree of freedom collective variable method is used to calculate
analytically the kink phase shift and the kink critical velocity sufficient to
pass through the defect. Comparison with the numerical results suggests that
the collective variable method is able to predict these parameters with a high
accuracy.
