The card itself, the root of the card, is writable (both the internal "card" and the external card). An app should be able to create a folder on the card. Then that app, and only that app (or root) can write anything in that folder (or if the group the app is a member of has write permissions, any app in that group can).
All KitKat changed was that the external SD card is now handled as normal Linux storage, not as the wild west of storage (where any app could write anything anywhere - no app could do that in internal storage before).
If you uninstall the app, delete its folders (use Settings/General/Storage/Miscellaneous - look for the app, press the box to check the app and press the trash can), install the app from scratch (from the Play Store, not from a backup, so you're getting the latest version) and it still can't write to its own folder, either the app isn't written to run on KitKat (it's trying to write to a folder it has no permission to write to) or the phone has a major problem (probably one that can be fixed by reflashing the ROM (which your carrier should be able to do, but I'd save that as a last desperate attempt - check with the app author first).