I tried my own version of (see below) and to my suprise, it didn't work! lol.

links returns an array of objects of type WWW::Mechanize::Link, not an array of strings.
foreach ($mech->links()) { print "$_<br>" }
should be
foreach ($mech->links()) { print $_->url(), "<br>" }

Is there a way I can check this link?

Is there any reason to believe it's not? Why are you assuming the module isn't working? WWW::Mechanize::Link has a method called text if you can get the link as an object of that type.