if ($mypath =~ m{(?:^|/)mydirectory/dirOne/pagehere\.cgi$})
{
   print "results here";
}

if ($mypath =~ m{(?:^|/)mydirectory/dirOne/[^/]+\.cgi$})
{
   print "results here";
}
[download]

(?:^|/)mydirectory
makes sure we don't match
notmydirectory

[^/]+
means "one or more of (not a slash)".