Thanks for that Toivo.
I've (eyeball) checked index.php, as well as defines.php and framework.php in the includes directory, they're all the same as those in the full download package (I downloaded that to check, although the site had been updated from the update package).
My (not particularly Joomla aware) hosting company says
We have looked into this there doesn't look to be evidence of a hack and the errors themselves are coming from the sendmail wrapper on the hosting server when the site is sending mail, likely due to something wrong with the email being sent, for example a missing header or bad formatting.
I don't know how the site could be sending email with something wrong unless something malicious was happening. Its only sending function is currently the "contact us" page. We haven't received any mail. The site is set to use phpMail. I don't know enough to know if that uses the file that's complaining, but sending a test message succeeds easily enough.
The times of the errors are a little suspicious, broadly similar over three days.
It's a community, the more we all contribute, the better it will be.