/wp-includes/plugin.php error – function not found or invalid function name

Some time ago we started seeing various error messages at the top of the dashboard which said something about functions in plugin files.

The usual error message is:
Warning: call_user_func_array() expects parameter 1 to be a valid callback, function ‘admin_init’ not found or invalid function name in /homepages/31/d575245533/htdocs/data-broker.co.uk/wp-includes/plugin.php on line 503

The dashboard has very reduced functionality – we can’t search, see content or make changes, if we do we are taken to a blank page with the following error messages:
Warning: call_user_func_array() expects parameter 1 to be a valid callback, function ‘admin_init’ not found or invalid function name in /homepages/31/d575245533/htdocs/data-broker.co.uk/wp-includes/plugin.php on line 503

Warning: Cannot modify header information – headers already sent by (output started at /homepages/31/d575245533/htdocs/data-broker.co.uk/wp-includes/plugin.php:503) in /homepages/31/d575245533/htdocs/data-broker.co.uk/wp-includes/pluggable.php on line 1207

Warning: Cannot modify header information – headers already sent by (output started at /homepages/31/d575245533/htdocs/data-broker.co.uk/wp-includes/plugin.php:503) in /homepages/31/d575245533/htdocs/data-broker.co.uk/wp-includes/pluggable.php on line 1207

I’ve tried disabling all our plugins one by one and even with all plugins disabled it makes no difference.

All plugins, wordpress versions and databases are up to date.