1. Make sure that ...\cygwin\bin is in your PATH.
2. Modify your "latex-workshop.latex.tools" configuration, or specifically the latexmk tool, as follows:

{
    "latex-workshop.latex.tools": [
      {
        "name": "latexmk",
        "command": "bash",
        "args": [
          "-c",
          "latexmk -synctex=1 -interaction=nonstopmode -file-line-error -pdf -outdir=\"`cygpath \"%OUTDIR%\"`\" \"`cygpath -u \"%DOC%\"`\" | sed 's#/cygdrive/\\([a-zA-Z]\\)/#\\1:/#g' && (gunzip -kf %DOCFILE%.synctex.gz && sed 's#/cygdrive/\\([a-zA-Z]\\)/#\\1:/#g' <%DOCFILE%.synctex | gzip -c - >%DOCFILE%.synctex.gz ; rm -f %DOCFILE%.synctex)"
        ],
        "env": {}
      }
    ]
}

Description of what's going on:

• -outdir=\"cygpath "%OUTDIR%"\" specifies the output directory while mapping the path into Cygwin naming
• \"cygpath -u "%DOC%"\" specifies the input file while mapping the path into Cygwin naming
• | sed 's#/cygdrive/\\([a-zA-Z]\\)/#\\1:/#g' converts the output to use Windows naming, so that LaTeX Workshop can parse it. For example, this enables clicking on error messages to go to the correct line.
• (gunzip -kf %DOCFILE%.synctex.gz && sed 's#/cygdrive/\\([a-zA-Z]\\)/#\\1:/#g' <%DOCFILE%.synctex | gzip -c - >%DOCFILE%.synctex.gz ; rm -f %DOCFILE%.synctex) converts the synctex file to use Windows naming (and ignores error messages), so that you can jump back and forth between corresponding parts of the PDF and TeX files.