The thing about the Hot questions widget is that it's a neat idea, and well executed. However, it shows questions from all around the network. And most of the time from topics I have no interest in.
I suggest another widget be added* to show questions from a specific user-defined set of sites. There could also be the choice of showing "new" questions** or "hot" questions from the chosen sites.
I envision this new widget to be useful as it will give us hot trending questions from our favorite sites to read on (less refreshing, more reading) or new questions, encouraging us to contribute more to the site.
Also, tapping the site icon should open the relevant list of questions at the site, not the current question. That will be a handy shortcut (can be applied to the existing widget as well).
* Or the existing widget could have added options to choose between hot network questions and the above.
** There could be an optional filter of selecting only new questions with 0 or more nett vote score for the widget.