plotting – Turn a region and points inside that region into a mesh

I have a region defined by a polygon and a few points that are inside that region. I’d like to create a mesh where its boundary is the boundary of the region and the vertices interior to the mesh are the points that I created (or more if needed for compatibility). How can I create a mesh like this?

region = Polygon({{0, 0}, {1, 0}, {1, -1}, {2, -1}, {2, 0}, {3, 
   0}, {3, -2}, {0, -2}});
points = RandomPoint(region, 1000);
mesh = createMeshFromRegionAndPoints(region, points);  (* This is the function I need to create *)

I know I can create a mesh from the region with DiscretizeRegion, but this won’t capture the interior vertices. I can also create a DelaunayMesh of the interior points, but given my region is non-convex, the boundary will be exceeded. If I could somehow “trim” the edges of the DelaunayMesh using the region, then that would get me most of what I want (I’m trying to make a ListDensityPlot that draws on my region).