It's actually not that hard. I missed in your example HTML that the button is just "decoration" for the a element (which is missing a closing quote after https://something), so you don't need to "click" a button, you need to follow a link.

Note that $url should be provided by someone.

use WWW::Mechanize;

$ua = WWW::Mechanize->new;

my $response = $ua->get($url);
my $next_response = $ua->follow_link(text_regex => qr/continue.../);
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I missed in your example that the button is just "decoration" for the a element (which is missing a closing quote after https://something), so you don't need to "click" a button, you need to follow a link.