Hi Gemma,
Thanks for the tracker link. It really helped. I've added some new information to the tracker issue and suggested a course of action to fix it. I suspect that this issue has existed since several years. I've come across posts dating back to 2015.
Based on my analysis, the issue only happens if you ever try to change the shortname of a profile field because the code, which enables you to change the profile field, doesn't reflect the change by updating the name field in the mdl_config_plugins table.
The easiest workaround is to rename the shortname back to its original value however this might affect some of your integration. Alternatively you can rename the values in the name fields of the mdl_config_plugins table to reflect the current shortnames.
Once you have it working again, don't rename any custom fields until this issue is resolved or be ready to apply the above mentioned workarounds.
in my case, it was triggered by upgrading to the 2018-05-31 release of Moodle 3.4.3 from a release just two weeks earlier. What did it was the fact that I renamed some custom fields in between. I was being prompted for 3 fields I had modified for each of the following authentication methods: CAS server (SSO), External database, LDAP server and Shibboleth.
Hope we can see a fix for this soon.
Michael