The common trick is to sort the words by length descending. The longer matches will then be tried first.

#! /usr/bin/perl
use warnings;
use strict;
use feature qw{ say };

my @files = qw( 20220401_note.txt
                20200101_no_match.txt
                20220303_page.txt
                20220101_page_with_blanks.txt
                20220111_page_blanks.txt );

my @get = qw( note page page_with_blanks page_blanks );
my $match = join '|',
            sort { length $b <=> length $a }
            map quotemeta,
            @get;

for my $file (@files) {
    if (my ($date, $name) = $file =~ /(202[0-9]{5})_($match)/) {
        say "D:$date - N:$name";
    }
}
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