I am developing an App for Firefox OS. The App is just my own website with some added functionality. What will be preferred to add new functionality to mobile website and make it Hosted Firefox OS App or creating a new packaged App?
Definitely go with the Hosted App in this case with emphasis on offline first. You already have a website so the domain name for your website can already be used to represent the mobile web app version of your site. There are tons of advantages for going the hosted web app route. I'll try to list some of them on that are on top of my head. By using a hosted web app you gain:
As a rule of thumb, if you can get away by doing a hosted app, I would always recommend it over the alternatives. Packaged and Privileged apps are essentially stop gaps until mobile web matures enough to supports all the APIs you would need. Hosted apps already support many APIs and if that is enough for your use case then you are set. Another way to look at it is that hosted apps are essentially the purest form of mobile web apps -- they are the closest thing to web applications that are accessible from mobile devices.
Make sure to do some research about responsive design and offline first.
Good luck with your Firefox and mobile web adventure!