For those having similar problems, the background seems to be that I'm using an e-mail redirection service for a domain that I own. I've been using this for many years, but the problems have only started recently, so I guess technology has caught up and now needs further authentication.
I reported this to the domain hosting company I use and they gave the following response:
I have added SPF and DKIM and DMARC records to your domain for you, this usually fixes any issues with e-mail deliverability issues such as the one you are having. Please allow 2-24 hours for this to fully propagate and please get back in touch if any issues reoccur. I will list the information that I have added below:
Domain Keys Identified Mail (DKIM) is a method for associating a domain name to an email message, thereby allowing a person, role, or organization to claim some responsibility for the message.
Sender Policy Framework (SPF) is an email validation system designed to prevent email spam by detecting email spoofing, a common vulnerability, by verifying sender IP addresses.
DMARC, (Domain-based Message Authentication, Reporting, and Conformance) is an email protocol; that when published for a domain; controls what happens if a message fails authentication tests (i.e. the recipient server can't verify that the message's sender is who they say they are).