Don't do it by hand, make a sub to do the quoting... or move to Net::OpenSSH that already has a shell quoting method (or just copy the relevant code from there ;-)

use Net::OpenSSH;

my $public_key = "...";

my $cmd1 = <<EOC;
uname -a
mkdir \$HOME/.ssh
chmod go-w \$HOME \$HOME/.ssh
echo "$public_key" >>\$HOME/.ssh/authorized_keys
EOC

$cmd1 =~ s/\n/;/g;
$cmd1 = Net::OpenSSH->shell_quote($cmd1);

my $cmd2 = '/usr/bin/ssh -t -l usr machineD ' . $cmd1;
$cmd2 = Net::OpenSSH->shell_quote($cmd2);

my $cmd3 = '/usr/bin/ssh -t -i $HOME/.ssh/LeadRNCkey -l usr machineC '
+ . $cmd2;

system $cmd3;
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