Plusnet mail may be accessed via the web browser, but when sending Plusnet uses your IP address to check you are online with Plusnet (by default, SMTP mail does not have any login mechanism, so spammers could send to anywhere if PN didn't check the IP address).
When you are using Three for your connection, you won't be considered a valid 'Plusnet' customer (as you are connecting to the internet via Three). If you use a web browser to login to your mailbox, then you will be able to send that way. I will see if there's any way to use the Plusnet mail service via my Three connection if you wish (but cannot promise it will be today).
I switched some years ago to Fastmail.net which means I can switch ISP without changing my mail address that friends and family know. For a small fee they allow you to send via their mail service (they have help pages to explain things, but I've talked someone through this over the phone to allow them to use my account to send mail when they had access via a firewall that only allowed web traffic (port 80 for web, other port numbers are used for POP SMTP and IMAP mail).