From what you have said so far, I can't be sure your problem is the same as mine. I'll explain my set-up, though, and leave it to you to decide if your situation is essentially the same.
I have a broadband account with Plusnet and email accounts with Apple (iCloud) and Google (Gmail). I have no problem sending emails from my iCloud and Gmail accounts on my Apple iMac via my Plusnet router. These are set up to use Plusnet's email servers. In addition, I have a Raspberry Pi which runs a personal website and the Easy Appointments booking system. By default, that uses the Linux sendmail daemon for email, which (if I've understood it correctly) acts as a local mail server. Emails sent from there would need the Plusnet email servers to act as an email relay, passing on any messages and bypassing any spam detection software used by Plusnet. But they don't allow that. In my case, the E!A emails were being discarded by Plusnet and I had to dig into the email logs to discover this.
In theory, I could have configured Easy Appointments to use one of my existing email accounts, but I couldn't find the server and port details easily and, anyway, it makes sense for my personal website to have its own email account. So I decided to create a new email account with my domain registrar. In the process, they gave me the server and port details I needed, and I entered them into the Easy Appointments application/config/email.php file.
It looks to me as though the message you are getting is telling you that Superspace (like Plusnet) doesn't allow you to use their email servers to relay messages from another server. I understand this is standard practice among email service providers these days. If I'm right, you will need to edit your application/config/email.php file so that Easy Appointments uses your Superspace email account instead of the default.
Hope this helps.