Maybe my experience with (or my use of) CGI::Application is different from yours, but for me, a typical url for executing "run_mode_x" via CGI::Application starts like this:

"someurl.pl/run_mode_x$params"
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where $params is either an empty string or else something like:

"?param1=value1(&...)"
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depending on whether or not the request needs to pass any parameter(s) to the chosen run_mode.