Actually you have to end every line of $smtp->datasend() with a ';' cause they are Different lines.
They are not part of the 'new' method
Also you should have your $to and your $from variables with the actuall email address, not what appears on the email.
So your code should look like this

my $to = "ghenry\@perl.me.uk";
my $from = "support\@perl.me.uk";
my $server = "relay.plus.net" ;
$smtp = Net::SMTP->new($server);
$smtp->mail($from);
$smtp->to($to);
$smtp->data();
$smtp->datasend("To: Gavin Henry <$to>\n");
$smtp->datasend("From: Support <$from>\n");
$smtp->datasend("Subject: Rsync backup success on $date\n");
$smtp->datasend("The remote backup has been completed.\n\n");
$smtp->datasend("Info:\n");
$smtp->datasend("\n$rsync\n");
$smtp->dataend();
$smtp->quit;
[download]

I also added a quit, to the smtp object, unless you intent to do anything else with it and the socket connection.