An assembly is provided for a turbine engine. This turbine engine assembly includes a combustor wall. The combustor wall includes a shell, a heat shield and an annular body. The annular body extends through the combustor wall and at least partially defines a quench aperture along a centerline through the combustor wall. The shell defines a first cooling aperture radially outwards of the annular body relative to the centerline and is configured to direct air to impinge against a portion of the annular body between the heat shield and the shell.