Detect and fix invalid polygon

I expect there's a simpler way to do this, but here is a possibility. FIrst, discretize the polygon, and then find the boundary:

boundary = RegionBoundary @ DiscretizeRegion @ poly

Simplify the boundary using an undocumented, internal function:

boundary = Region`Mesh`MergeCells @ boundary

Notice the defect is gone. Convert the output to a BoundaryMesh and extract the polygon:

simple = MeshPrimitives[
    BoundaryMeshRegion[MeshCoordinates[boundary], MeshCells[boundary, 1]],
    2
]   

{Polygon[{{2440.37, 163.438}, {2437.38, 166.855}, {2446.11, 174.504}, {2435.2, 183.216}, {2418.95, 168.237}, {2428.72, 160.067}, {2420.63, 153.885}, {2431.45, 142.989}, {2442.18, 152.113}, {2435.84, 159.581}}]}

The fixed polygon:

Graphics[{FaceForm[None], EdgeForm[Black], simple}]

And finally, here are the above steps packages as a function:

fixPolygon[poly_Polygon] := With[
    {boundary = Region`Mesh`MergeCells @ RegionBoundary @ DiscretizeRegion @ poly},

    MeshPrimitives[
        BoundaryMeshRegion[MeshCoordinates[boundary], MeshCells[boundary, 1]],
        2
    ]
]

Another example using a polygon from the comments:

fixPolygon @ Polygon @ {{3, 3}/2, {3, 3}, {0,1}, {3, 1}, {2, 2},{3, 3}/2}

{Polygon[{{3., 3.}, {0., 1.}, {3., 1.}, {2., 2.}}]}