Here's some hot takes for discussion.
Governments should use, develop and contribute to open-source software. Imagine instead of every country developing their own software systems for taxes, health records etc, there would be one or maybe a handful of solid implementations. This would reduce duplication of efforts globally and the open-source implementation, with its concentration of efforts, would probably be higher quality than the typical implementation currently. You might also view it as a form of aids for governments of poorer countries/states/cities who now have access to high-quality software that is probably much easier for them to put into practice. This is something that could be done at any level of government.
Instead of letting Silicon Valley constantly reinvent public transit cities should make an app where you can see where buses are currently, and shit like that.
911 emergency services should take advantage of all the extra functionality of smart phones. Perhaps you could have a system where mashing buttons on your phone triggers your phone to start streaming as much info as possible to 911 dispatchers in addition with putting you on a call. (Video from both cameras, GPS, accelerometer, mic etc). This might also be a boon for reliable evidence, especially in traumatic situations where first-hand testimony can be unreliable or imprecise.
The general theme I guess is that the public sector hasn't really fully taken advantage of the technology we have now while the private sector is throwing money at every dumb tech startup under the sun.