  We used polarization-resolved Raman scattering to study magneto-elastic
coupling in Ba(Fe$_{1-x}$Au$_{x}$)$_2$As$_2$ crystals as a function of light
Au-doping, materials for which temperatures of the structural transition
($T_S$) and of the magnetic ordering transition ($T_N$) split. We study the
appearance of the $A_g$(As)phonon intensity in the $XY$ scattering geometry
that is very weak just below $T_S$, but for which the intensity is
significantly enhanced below $T_N$. In addition, the $A_g$(As) phonon shows an
asymmetric line shape below $T_N$ and an anomalous linewidth broadening upon
Au-doping in the magnetic phase. We demonstrate that the anomalous behavior of
the $A_g$(As) phonon mode in the $XY$ scattering geometry can be consistently
described by a Fano model involving the $A_g$(As) phonon mode interacting with
the $B_{2g}$ symmetry-like magnetic continuum in which the magneto-elastic
coupling constant is proportional to the magnetic order parameter.
