How can I fix a nail pop in vinyl flooring?
A not-entirely satisfactory &quot;fix&quot; I've used:

Use a very sharp nail-set or strong awl to punch the nails well below the surface, being careful not to further damage the surface itself.
Fill the divot and cracks with a cement such as epoxy, removing loose pieces or those that cannot be pushed in. Leave the patch slightly proud (above) the surface.
Carefully sand the patch even.
Color the patch to blend with the flooring, perhaps with a furniture stain kit.
Overcoat with varnish, polish or wax.

The nails might pop up again, perhaps in as few years.