Thanks for the very quick reply. I’ve no custom code, but I did have old plugins and themes (although none of the plugins were active). However, I was very careful to follow the upgrade steps.
I was running version 6.4 because I couldn’t upgrade to 6.5 (see below).
Unfortunately, the problem prevents me from logging into the WP admin, so it’s not easy to test the issue. (I can log in if I rename the simplepress folder and therefore disable the plugin.) With SimplePress activated I did try renaming the sp-resources folder because all the themes etc. in there seemed very old, but that didn’t help.
After deactivating SimplePress I was then able to update it. Reactivating 6.5 did allow me into the admin, though. (Perhaps because I already had a log-on cookie?)
I was then able to update the SP plugins, although I had significant difficulty in doing so as I frequently got the message that the plugin was already at the latest version. I was doing them one at a time to see which might be causing the problem. I discovered that after updating one plugin from update-core.php I had to visit the SimplePress plugins page before I could update another. I then end I gave up doing them one at a time and just did them all together.
However, none of this helped. Nor did visiting the permalinks settings page.