I tried to do what you advised and it didn't work.
could you please check field you have unchecked
Why? In documentation
The System properties tab contains the general properties that all widgets and web parts have in common. Their default values can be modified by clearing the Inherited checkbox and entering the desired value into the appropriate field. All changes must be confirmed by clicking the OK button.
I want to have editable text widget, which will be wrapped by html I entered in Content before/after. Or by html from web part container. So for end user it will be like predefined html with editable text area (or html, or text box) inside.
It looks like it doesn't work right now so it could be that I'm doing something wrong, or there is some issue with Kentico, or both. Can you please provide instructions about how I can reach this?