If you have geostatistical analystGeostatistical Analyst you can use the Inverse Distance Weighting tool to fill in your gaps.
I think IDW is the most appropriate for your use-case of the various interpolation tools available..., though they do require Spatial Analyst license.
That outputsIt will output a raster... Then, you can then convert back to points at your desired resolution, then use Extract Values to Points to join back your z values. See: Assigning Raster Value to Point Data