Does anyone know the part number of the black piece that retains the square nut
for the fog light mount?

Is the square nut held in only by the black piece and is the black piece just plastic that's pressed into the body?

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The black is the remnants of a plastic or rubber washer. It should chip right off and fall away.

The nut is supposed to 'float'. The hole through the body is oversize, and the square nut is captivated from the back but a steel 'cage' that's spot welded to the inside.