Steps to reproduce the issue

Issue # 1:

Develop a module with a configuration field that uses com_ajax (e.g. to dynamically retrieve options based on other field values).
Install the module and create a module instance.
Disable the module extension.
Edit the module in backend.

Issue # 2:

Disable site's Feed Display module in Extension Manager.
Enter this URL: administrator/index.php?option=com_ajax&module=feed&format=json

Expected result

Issue # 1: Module can be configured normally.

Issue # 2: Fails with message:

Method getAjax does not exist

Actual result

Issue # 1: Ajax-based field doesn't work with error message:

Module mod_example is not published, you do not have access to it, or it's not assigned to the current menu item.

Issue # 2: Fails with message:

Module mod_feed is not published, you do not have access to it, or it's not assigned to the current menu item.

System information (as much as possible)

3.9.11-dev.

Additional comments

For issue # 1 I think we shouldn't block the use of Ajax when module is disabled. This should really be left up to the module based on its use case.

If issue # 1 is fixed, issue # 2 is fixed automatically. Otherwise, we need to implement client filter. But that should be based on URL parameter rather than on current client.