Colorize mesh according to the nodepoint colors

Do you mean something like this?

R = DelaunayMesh[p];
Graphics[
 GraphicsComplex[
  MeshCoordinates[R],
  MeshCells[R, 2, "Multicells" -> True],
  VertexColors -> c
  ]
 ]

Edit

Getting colorgradients on the edges only seems to be somewhat trickier as VertexCoordinate for Line objects in a GraphicsComplex does not work. A workaround could be this:

elist = Flatten[MeshCells[R, 1, "Multicells" -> True][[1, 1]]];
Graphics[
 Line[
  Partition[MeshCoordinates[R][[elist]], 2],
  VertexColors -> Partition[c[[elist]], 2]
  ]
 ]

SeedRandom[1]
p = RandomReal[{0, 1}, {10, 2}];
c = Map[Hue[#] &, RandomReal[{0, 1}, 10]];

1. Post-process MeshPrimitives to add VertexColors:

rule = pr : (_Polygon | _Line | _Point) :> 
  Append[pr, VertexColors -> 
    AssociationThread[p, c] /@ First[pr /. Point[x_] :> Point[{x}]]]

Row[Graphics[{AbsoluteThickness[5], AbsolutePointSize[12], 
     MeshPrimitives[DelaunayMesh[p], #] /. rule,
     AbsoluteThickness[3], Black, Circle[#, Offset[6]] & /@ p}, 
    ImageSize -> 300, PlotLabel -> Style[#, 16, Black]] & /@
 {0 | 1, 0 | 2, 0 | 1 | 2}, Spacer[5]]

2. Use the option MeshCellShapeFunction with desired styles/primitives for vertices, edges and faces:

Row[{DelaunayMesh[p, ImageSize -> 400,
   MeshCellStyle -> White,
   MeshCellShapeFunction ->
    {{0, All} -> ({c[[#3[[1, 1]]]], Disk[#, Offset[7]]} &),
     {1, All} -> ({AbsoluteThickness[7], CapForm["Round"], 
         Line[#, VertexColors -> c[[#3[[1, 1]]]]]} &),
     {2, All} -> None}],
  DelaunayMesh[p, ImageSize -> 400,
   MeshCellStyle -> White,
   MeshCellShapeFunction ->
    {{0, All} -> ({c[[#3[[1, 1]]]], Disk[#, Offset[7]]} &),
     {1, All} -> None,
     {2, All} -> ({EdgeForm[Gray], 
         Polygon[#, VertexColors -> c[[#3[[1, 1]]]]]} &)}]}]

If you're interested in doing this with 3D mesh objects (as I was) the following works in Mathematica 12.1:

mesh = BoundaryDiscretizeRegion[Ball[{0, 0, 0}, 1], 
  MaxCellMeasure -> {"Length" -> 1}, PrecisionGoal -> 0.01] (* or any mesh object *)
vertexscalar = RandomReal[{0, 1}, Length[MeshCoordinates[mesh]]]

Graphics3D[
 GraphicsComplex[MeshCoordinates[mesh], 
  Polygon[MeshCells[mesh, 2][[All, 1]]], 
  VertexColors -> ColorData["DarkRainbow"] /@ Rescale[vertexscalar]]]