User with permission can't add an event to calendar by clicking
Can you clarify? You said:


  users, who are allowed to add an event, can't do it. They can add an
  event by navigation and clicking on"add content", but they can't do it



what do you mean they can't do it? Can they navigate to /node/add/[event] (I'm not sure what your event node type is called)? Or when they do go to /node/add/[event] does something happen when they try to save the new event?


Are you sure you have the permissions set correctly for their role? Check out the permissions at admin/people/permissions. Double check the permissions under Node.