To avoid hard, jagged halftone edges when processing pixel art, you can export or rescale your art to a larger scale factor before loading in into the shader- and then set the texture filtering to "linear" or "cubic"! The scale depends on the size of the art and the desired render size, try 600-800%.

This will smooth out the hard halftone pattern edges! :)

Do note that, at higher resolutions and higher halftone scales, this may not be necessary, as the halftone dots will be too small for the hard edges to be noticeable! It all depends on the rendering scale (and intent).