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 body extends laterally between an inner surface and an outer surface. The inner surface defines an igniter aperture in the combustor wall. The outer surface is vertically between the heat shield and the shell. The shell defines a first cooling aperture through which air is directed to impinge against the outer surface.