Three different scenarios and outcomes, but they seem to be related, so I put them together.

Scenario 1:

1. In Dashboard enter edit mode and add a Clock widget (don't leave the edit mode);
2. In a new browser tab disable the Clock Dashboard Widget Module (Administration -> General -> Modules);
3. In Dashboard try to add another widget.

Result (1):
There is an error in the Add widget modal. This is likely because when adding a new widget the last choice is remembered. The user is not able to add another widget:
Expected (1):
A default widget type is selected instead.

Scenario 2:

1. In Dashboard enter edit mode and add a Clock widget (don't leave the edit mode);
2. In a new browser tab disable the Clock Dashboard Widget Module (Administration -> General -> Modules);
3. In Dashboard press the [Save changes] button.

Result (2):

Dashboard is not saved and an error is shown instead:

Expected (2):

Either save this Widget and show it as an unavailable Widget, or maybe just leave as is, but show the widget type in the error message, so the user knows what to delete.

Scenario 3:

1. Disable the Clock Dashboard Widget Module (Administration -> General -> Modules);
2. In a new browser tab open Dashboard, enter edit mode (don't leave the edit mode);
3. Enable the Clock Dashboard Widget Module;
4. In Dashboard add a new widget and choose Clock, press [Add].

Result (3):

The widget does not get added, also there is no feedback to the user as to why:

Expected (3):

Either the widget gets added, or there is an error message.