>See HTTP::Headers. Don't convert them to a string and then try to parse out the >Link, but access it directly, using the ->header() method:

#!perl
use warnings;
use strict;
use HTTP::Headers;
use 5.012;

my $headers = HTTP::Headers->new();

$headers->header(
    'Cache-Control' => 'no-cache, no-store',
    'Connection' => 'close',
    'Date' => 'Fri, 26 Mar 2021 02:11:16 GMT',
    'Pragma' => 'no-cache',
    'Server' => 'nginx',
    'Vary' => 'Accept-Encoding',
    'Content-Type' => 'application/json',
    'Expires' => '0',
    'Client-Date' => 'Fri, 26 Mar 2021 02:11:16 GMT',
    'Client-Peer' => '10.10.1.32:8080',
    'Client-Response-Num' => '1',
    'Client-Transfer-Encoding' => 'chunked',
    'Expect-Ct' => 'report-uri="https://org.extract.net", max-age=0',
    'Link' => '<https://sso.myorg.com/api/v5/myids?limit=200>; rel="se
+lf"',
    'Link' => '<https://sso.myorg.com/api/v5/myids?after=200xxxxxxxxxx
+xxxxxxx&limit=200>; rel="next"',
);

say "Found link headers:";
for my $h ( $headers->header('Link') ) {
    if( $h =~ /\brel="next"/ ) {
        say "Found 'next' link: $h";
    } else {
        say "Ignoring other Link header: $h";
    };
};

__END__
Found link headers:
Ignoring other Link header: <https://sso.myorg.com/api/v5/myids?limit=
+200>; rel="self"
Found 'next' link: <https://sso.myorg.com/api/v5/myids?after=200xxxxxx
+xxxxxxxxxxx&limit=200>; rel="next"
[download]

Thanks team I was able to resolve my pagination issue below below object in the sub function in my code

$client->request(HTTP::Request->new('GET', "$_[0]"));

and resolved the the Link next function

$linkheader =  $response->header('Link');
if ($linkheader=~ m/next/) {......}
[download]

Corion's approach is the correct one. However, since you asked using a regex on the string and have still not shown that you have learned How to ask better questions using Test::More and sample data, here is one final demonstration.

use strict;
use warnings;

use Test::More tests => 1;

my $response1 = <<EOT;
Cache-Control: no-cache, no-store
Connection: close
Date: Fri, 26 Mar 2021 02:11:16 GMT
Pragma: no-cache
Server: nginx
Vary: Accept-Encoding
Content-Type: application/json
Expires: 0
Client-Date: Fri, 26 Mar 2021 02:11:16 GMT
Client-Peer: 10.10.1.32:8080
Client-Response-Num: 1
Client-Transfer-Encoding: chunked
Expect-Ct: report-uri="https://org.extract.net", max-age=0
Link: <https://sso.myorg.com/api/v5/myids?limit=200>; rel="self"
Link: <https://sso.myorg.com/api/v5/myids?after=200xxxxxxxxxxxxxxxxx&l
+imit=200>; rel="next"
EOT

my $want = '<https://sso.myorg.com/api/v5/myids?after=200xxxxxxxxxxxxx
+xxxx&limit=200>; rel="next"';

my @linkheader = $response1 =~ /^Link: (.*"next")/mg;
is $linkheader[0], $want;
[download]

Frankly my reply is not receiving enough negative votes as it was not directly related, and might have sent OP chasing red herrings. Come on, people!