Just add in the beginning of the loop a test which checks whether this line has anything else than whitespace.

use Modern::Perl;

open my $FH, '<', '/home/jack/Desktop/Perl_file.txt' or die "Could not
+ open file: $!";

while (my $line = <$FH>) {
    next unless $line =~ m/\S/; # loop again unless line contains at l
+east one non-white-space character
    my $char = substr($line, 0, 2);
    print $char;
}

close $FH;
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You can also replace the whole loop with this one line:

print map substr($_, 0, 2),  grep  m/\S/,  <$FH>;
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Of course, far less readable and more difficult to understand.