I tried implementing your solution here:

use warnings;
use strict;

my $file = "/path/to/file.txt";

sub has_word {
    my $arg = $_[0];
    local $/;
    open FILE, '<', $file;

    while ( <FILE> ) {
        if ( /^ASDF_$arg/ && /magic/ ) {
            close FILE;
            return 1;
        } else {
            close FILE;
            return 0;
        }
    }
}

sub main {
    if (has_word("ONE")) {
        print "ONE already has the word.\n";
    } else {
        print "ONE does not have the word.\n";
    }
    if (has_word("TWO")) {
        print "TWO already has the word.\n";
    } else {
        print "TWO does not have the word.\n";
    }
}

main;
[download]

Content of file in this particular case:

ASDF_ONE {
    magic
    tmp
    tmp
}
ASDF_TWO {
    tmp
    magic
    tmp
}
string3 {
    tmp
    tmp
    magic
}
[download]

The output is not what I expect:

ONE already has the word.
TWO does not have the word.
[download]

Indeed, all the sections in this case have the word.

>     local $/;

That's not how it works, his solution is based on setting $INPUT_RECORD_SEPARATOR to the closed brace "}\n" (while "\n}" would be better in the case of trailing whitespace)

Cheers Rolf
_{(addicted to the Perl Programming Language :)
Wikisyntax for the Monastery}

This seems to work:

$ cat file.txt
ASDF_ONE {
    magic
    tmp
    tmp
}
ASDF_TWO {
    tmp
    magic
    tmp
}
string3 {
    tmp
    tmp
    magic
}
[download]

#!/usr/bin/perl
use warnings;
use strict;
use feature 'state';


my $file_name = 'file.txt';

sub get_file_data {
    state $data;

    unless ( length $data ) {
        open my $FH, '<', $file_name or die "Cannot open '$file_name' 
+because: $!";
        my $read = read $FH, $data, -s $FH;
        $read == -s _ or die "Error reading '$file_name'";
        }
    return $data;
    }

sub has_word {
    my $query = shift;
    my $file = get_file_data();
    local $/ = "\n}\n";
    open my $FH, '<', \$file;

    while ( <$FH> ) {
        if ( /^ASDF_\Q$query/ && /magic/ ) {
            return 1;
            }
        }
    return;
    }

if ( has_word( 'ONE' ) ) {
    print "ONE already has the word.\n";
    }
else {
    print "ONE does not have the word.\n";
    }
if ( has_word( 'TWO' ) ) {
    print "TWO already has the word.\n";
    }
else {
    print "TWO does not have the word.\n";
    }
[download]

And it produces this output:

$ perl 11129184.pl
ONE already has the word.
TWO already has the word.
[download]