There is no way to remove the nails that will not leave a jagged hole.

A cementious underlayment patch will do the job. Look for brand names, such as Ardex Feather Finish (sometimes under the Henry's or Armstrong labeling) or Mapei's Planipatch or an equivalent product. Apply the patch, wait for it to set and apply another, if necessary, to bring the filled area flush with the terrazzo surface.

Don't use a gypsum-based patch.