Following the constructive feedback here is the working code:

#!/bin/perl

use strict;
use warnings;


my $input_file = 'input.txt';
my $output_file = 'output.txt';


open(my $in_file,'<', $input_file) or die "Can not open file $input_fi
+le for writing: $!.\n";
open(my $out_file,'>', $output_file) or die "Can not open file $output
+_file for writing: $!.\n";

if (not (grep{/Monday/} <$in_file>) {

    seek $in_file, 0, 0;
    $. = 0;

    while (<$in_file>) {
        print $out_file $_;

        if ($. == 1) {
            print $out_file "Friday\n";
        }
    }
}

close($in_file);
close($out_file);
[download]

The same code but with 'unless' instead of 'if (not)'

#!/bin/perl

use strict;
use warnings;


my $input_file = 'input.txt';
my $output_file = 'output.txt';


open(my $in_file,'<', $input_file) or die "Can not open file $input_fi
+le for writing: $!.\n";
open(my $out_file,'>', $output_file) or die "Can not open file $output
+_file for writing: $!.\n";

unless (grep{/Monday/} <$in_file>) {

    seek $in_file, 0, 0;
    $. = 0;

    while (<$in_file>) {
        print $out_file $_;

        if ($. == 1) {
            print $out_file "Friday\n";
        }
    }
}

close($in_file);
close($out_file);
[download]

As 'seek' could probably be avoided if I rethink things in a different way, I'll get back to this when I have clarified my thoughts

I appriciated