My ultimate gall here is to search for the string 'Friday' in the file 'input.txt', if 'Friday' is missing, add'Friday' to 'input.txt on line 2.

Just do it! But first a code review.

#!/bin/perl

use strict;
use warnings;


my $input_file = 'input.txt';


open(my $in_file,'<',$input_file) or die "Can not open file $input_fil
+e for reading: $!.\n";

if (grep{/Friday/} $in_file) { # WRONG. You are grepping the filehandl
+e,
                               # not the content of the file.
    print "Match\n";
} else {
    print "No match\n";
}

print "Before\n";             # "Before" is long ago.

while (<$in_file>) {
    print;                    # you don't look for "Friday" looping li
+nes.
}

print "After\n";              # after "After" nothing happens but file
+ closing.

close($in_file);
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A filehandle is a filehandle, not the content of the file. In the while-loop you read the file line-wise, but you don't do anything but printing what you've read. Also, you don't look for line number 2. The line number read is stored in the perl special variable $. which you should test, if you want to add (append?) the string "Friday" to line two. Fix:

#!/bin/perl

use strict;
use warnings;

my $input_file = 'input.txt';

open(my $in_file,'<',$input_file) or die "Can not open file $input_fil
+e for reading: $!.\n";

while (<$in_file>) {
    if (/Friday/) {
        warn "match on line $.\n"; # warn goes to STDERR
    } elsif ($. == 2) {            # no match on line 2
        chomp;                     # remove line ending
        $_ .= "Friday\n";          # add "Friday" and line ending
        warn "added 'Friday' to line $.\n";
    }
    print;
}

close($in_file);
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If you want inplace edit, you can place the directives for that after the first line of the script and omit the file opening and the while()-loop as well as the print statement. See perlrun. Read that.
This edits your 'input.txt' file in-place, moving the original to 'input.txt.bak' first:

#!/bin/perl -l -pi.bak

use strict;
use warnings;

if (/Friday/) {
    warn "match on line $.\n";          # warn goes to STDERR
} elsif ($. == 2) {                     # no match but on line 2
    $_ .= "Friday";                     # add "Friday" (and line endin
+g via -l)
    warn "added 'Friday' to line $.\n"; # warn doesn't add line ending
+ via -l
}
[download]

Hope that helps (HTH).