If the general emails are not IMAP and/or on this server, then do not use an IMAP forwarder. Instead, create a forwarder with their email provider.
- Go to the server which the site sits on.
- Go to account information
- Click List accounts
- Search for the domain you wish to use
- Click the orange CP logo
- In the email section, click Forwarders
- Click add forwarder
- Enter the email address to forward
- Make sure the domain is correct
- Enter the destination email address
- Click Add Forwarder
- Go back to cPanel tools, and ensure Email Routing is set to Local.