In order to combat the spread of spam, two of the techniques employed by (some) ISPs and SMTP providers are:
1) To block port 25 (SMTP port) outbound from any device on the ISP's network to any server other than the recognised SMTP server the ISP provides for it's customers
2) To require customers to log into the SMTP server they are using to send mail through with login credentials
It sounds like Fasthosts are eschewing the blame of you not being able to talk to their provided SMTP server (if indeed they do provide one) by offloading the support headache to the relevant ISP. It's a semi-decent enough practice even if it does circumvent them actually doing any work on mail delivery troubleshooting.
It also sounds like your SMTP outbound via your GMail account is using the settings with which you log into your GMail account to retrieve mail in order to authenticate you when you send outbound email via that connection.
Therefore...your solutions are:
A) To provide the smtp.mail.yahoo.com connection with a relevant set of credentials that can authenticate you to that SMTP server
or
B) To use the GMail SMTP setup and associated authentication credentials in your setup configuration for your Fasthosts account
HTH.
Mike.