Apple - Can I reinstall a single iOS app without losing its data?

While I never tried it myself, in theory you could use something like PhoneDisk to open the app's private folder and copy all the content to your computer. Then, after deleting and reinstalling the app, you should be able to copy the content back.

Xcode can reinstall an app. Go to Window -> Devices and click on your device on the left. You can tap the '+' button and browse for an app, or just drag any app you want onto the 'Installed apps' section. You can even drag an app icon straight from iTunes. The app will be reinstalled, over the old one, without erasing any data, documents, caches, etc.

Reinstalling the application will almost certainly not solve your problem. iOS applications are digitally signed. If even a single byte is changed, iOS will refuse to run it. What you will be reinstalling will be identical to what you currently have installed.

The problem is far more likely to be with your data or the interpretation of it. There's probably a bug in the application that has either deleted your data or fails to recognise it. You should contact the developers to try to diagnose what is happening here.

You can get your data off your device by using a tool like iExplorer, or, if the application has enabled it, iTunes File Sharing (which is available in the Apps tab for your device in iTunes).