A Smooth and Round Voronoi Mesh

1 + 4

We can discretize the rounded Polygon objects and then add the negative of the mesh through Prolog.

rm = DiscretizeGraphics[roundedPolygon[#, 0.3] & /@ MeshPrimitives[mesh, 2]]

Now there's some floating point differences in the results from roundedPolygon that seem to effect subsequent Boolean operations. We can fix this crudely merging nearby points.

coordsnew = Mean /@ Nearest[MeshCoordinates[rm], MeshCoordinates[rm], {All, 10^-12.}];
rm = MeshRegion[coordsnew, MeshCells[rm, 2]];

Now find the difference:

diff = BoundaryMesh @ RegionDifference[
  Cuboid @@ Transpose[CoordinateBounds[MeshCoordinates[rm], Scaled[.05]]], rm]

And assemble:

joints = With[{comps = ConnectedMeshComponents[diff]},
   If[Length[comps] == 1,
    {},
    Show[
      BoundaryMeshRegion[
       RegionUnion[Rest[SortBy[comps, RegionBounds]]], 
       MeshCellStyle -> {1 -> None, 2 -> GrayLevel[.3]}]
    ][[1]]
   ]
 ];

MeshRegion[
  rm, 
  MeshCellStyle -> {1 -> {Thick, GrayLevel[.3]}, 2 -> LightBlue}, 
  Prolog -> joints
]

3

It seems roundedPolygon is effected by an unnecessary run of consecutive duplicate points. We can fix this by deleting them.

roundedPolygon[p:Polygon[_?MatrixQ], zero_?PossibleZeroQ, ___] := p

roundedPolygon[Polygon[opts_?MatrixQ], r_?Positive, n : (_Integer?Positive) : 12] := 
  With[{pts = Split[opts][[All, 1]]},
    Polygon[Flatten[arcgen[#, r, n] & /@ 
     Partition[
      If[TrueQ[First[pts] == Last[pts]], Most, Identity][pts], 
      3, 1, {2, -2}
     ], 1]]
  ]

---

Edit

We can use MeshCellShapeFunction to preserve the data in the original mesh while having custom rounded cells:

meshsty = MeshRegion[
  mesh, 
  MeshCellShapeFunction -> {2 -> (roundedPolygon[Polygon[#], 0.3]&)}, 
  MeshCellStyle -> {1 -> {Thick, GrayLevel[.3]}, 2 -> LightBlue}, 
  Epilog -> joints
]

Notice that this only the visualization is affected and not the underlying data:

RegionEqual[mesh, meshsty]

True

Whereas the original solution does change the underlying data:

RegionEqual[mesh, rm]

False

1. fill the gaps between cells

Graphics[{PointSize[1 / L2 / 3], Red, MeshPrimitives[mesh, {0, "Interior"}], 
  {Directive[LightBlue, EdgeForm[Gray], EdgeThickness -> .001], 
    roundedPolygon[#, 0.3]} & /@ MeshPrimitives[mesh, 2]}]