Thanks for visiting the Apple Support Communities. From your post, I understand that you're having trouble with the weather widget. Specifically, it does not seem to be updating the weather as expected. I'm happy to help.
Since restarting does help temporarily, but not permanently, the next suggestion I have is to make sure your iOS software is up to date: Update your iPhone, iPad, or iPod touch - Apple Support .
If the iOS software is up to date and the issue continues, check your date and time settings to make sure they are accurate: How to get updated time-zone information for your Apple device - Apple Support .
I can also suggest removing, and readding the widget: Use widgets on your iPhone, iPad, and iPod touch - Apple Support .
Have a great day!