Creating a Hexagonal Lattice with VoronoiMesh

Here is what I was suggesting in comments:

SeedRandom[]

relaxed = Nest[
   PropertyValue[{VoronoiMesh[#, {{-1, 1}, {-1, 1}}], {2, All}}, MeshCellCentroid] &,
   RandomReal[{-1, 1}, {45, 2}],
   500
];
mesh = VoronoiMesh[relaxed, {{-1, 1}, {-1, 1}}, MeshCellStyle -> {1 -> White}];

Then extract the cell primitives corresponding to the interior cells and generate a new Mesh object:

interiorMesh = MeshRegion[
   MeshCoordinates[mesh],
   MeshCells[mesh, {2, "Interior"}],
   MeshCellStyle -> {1 -> White}
 ]

---

Depending on whether the application focuses on visualization or further computation, one could prefer having the output as simple Graphics objects instead, which are far easier to style than mesh components, at least for me:

Graphics[{
  Darker@Green, EdgeForm[{Thick, White}], 
  MeshPrimitives[mesh, {2, "Interior"}]
}]

If you are not completely attached to Voronoi, you might consider tiling with hexagons and then perturbing their coordinates. GraphicsComplex makes it work.

Define a hexagon.

HexTile[s_] := Polygon[s*{{Sqrt[3], 1}/2, {0, 1}, {-Sqrt[3], 1}/2,
                          {-Sqrt[3], -1}/2, {0, -1}, {Sqrt[3], -1}/2}]

Allow for translation.

TranslateObject[p_, {x_, y_}] := Map[{x, y} + # &, p, {2}]

Make a grid of hexagons.

HexGrid[s_, h_, v_] :=
   Flatten[Table[
      TranslateObject[HexTile[s], s {i*Sqrt[3] + Mod[j, 2]*Sqrt[3]/2, 3 j/2}],
      {i, 0, h}, {j, 0, v}], 1]

Make a perturbed grid of hexagons.

HexGridPerturbed[s_, h_, v_, r_] := 
   Block[{poly = Map[Round[#, 10.^-10] &, HexGrid[N[s], h, v], {2}], p, m, rules},
      p = DeleteDuplicates[Flatten[poly[[All, 1]], 1]];
      m = Length[p];
      rules = Dispatch[Thread[p -> Range[m]]];
      GraphicsComplex[
         p + RandomReal[{-r, r}, {m, 2}],
         poly /. rules]
   ]

Manipulate

Manipulate[
   Graphics[{
      EdgeForm[{Thick, White}],
      HexGridPerturbed[s, h, v, r]}],
   {{s, 1, "Hexagon Size"}, 0.1, 3., Appearance -> "Labeled"},
   {{h, 5, "Horizontal Count"}, 1, 10, 1, Appearance -> "Labeled"},
   {{v, 3, "Vertical Count"}, 1, 10, 1, Appearance -> "Labeled"},
   {{r, 0., "Random Noise"}, 0., 1., Appearance -> "Labeled"}
]

Comment:

With this code, I was able to make a 3D impression saving the result of the code in a file with the extension STL.

mesh = VoronoiMesh[pts]; 
hexagons = Select[Length[#[[1]]] == 6 &] @ MeshPrimitives[mesh, {2, "Interior"}]; 

DiscretizeGraphics @ Graphics @ hexagons

SeedRandom[1]
rt = 0.5;
pts = Flatten[Table[{3/2 i + RandomReal[rt], 
  Sqrt[3] j + Mod[i, 2] Sqrt[3]/2 + RandomReal[rt]}, {i, L + 2}, {j, L + 2}], 1];
mesh = VoronoiMesh[pts];

hexagons = Select[Length[#[[1]]] == 6 &] @ MeshPrimitives[mesh, {2, "Interior"}]; 
DiscretizeGraphics @ Graphics @ hexagons