Thanks very much salva, your advice was brilliant. Got it working now, you've saved me days of frustration! For anyone else in my situation who is new to this stuff, I fixed this by adding the ctl_dir option:  my $ssh = Net::OpenSSH->new($host, user=>$user, ctl_dir=>$ctldir);. I created the $ctldir (called .libnet-openssh-perl) within my /var/www directory and ensured the owner was www-data.
I hope this can help someone else too! Thanks again salva, legend!

Genius! Thanks very much! Is there anything special I need to do to get this to work when running the script as www-data? On the command line (to emulate running the webform) I run it as follows:  sudo -u www-data perl -T test but get an error message: Can't ssh to host: unable to create ctl_dir /var/www/.libnet-openssh-perl
This might be more of a general Linux question, in which case I apologise. I am able to ssh as the www-data outwith the perl script though, so I don't know what is wrong.
Thanks very much for your advice, you seem to know a lot about using this module and I'm very, very new to this sort of thing.

The module is trying to create the directory /var/www/.libnet-openssh-perl but it can't. It needs it to place a UNIX socket there for communication with the OpenSSH client.

The module documentation has a TROUBLESHOOTING section that discusses most of the issues you can encounter in detail. You should read it. But in any case, fell free to ask here again if that doesn't solve your problems.

you seem to know a lot about using this module

Hey, I should! I wrote it!