It is not currently possible to customize the 'My Tasks' component on the Home Page Layout.
If you'd like to see a change to this functionality, we suggest that you review, vote, and comment on the Custom Activity list view Idea on the IdeaExchange. If it doesn't cover the functionality you'd like to see, consider posting your own Idea. The product team monitors the idea exchange, and enhancements have been made due to customer feedback - see: Modify Home Page 'Tasks' Views).
As a workaround, you can create a Custom Link to a Task List View:
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