Excluding points from 2d plot

I am trying to plot certain regions using ContourPlot and RegionPlot in two dimensions. From these implicitly defined contours I need to exclude certain points. In the documentation of Exclusions it says that for a domain defined by $ n$ variables, exclusions must be of dimension $ n-1$ . Since points are dimension 0 objects, it does not work to exclude them naively. My idea to circumvent this issue was to include short line segments through the points and exclude the lines instead, which would not be a problem with my plot. However, I do not find anywhere how this can be done. An example for a plot can be found in this question. Any help is much appreciated.