Does the app show battery charge cycles and health information?

Unfortunately, this information is not accessible through the iOS API. In the past, some apps managed to use the private API (which is explicitly prohibited by Apple) and sneak through the review process onto the App Store, or used various hacks to get this information, such as the techniques described here:

https://stackoverflow.com/questions/36024247/getting-battery-health-information-on-ios

None of these seem to work now.

Even if some apps in the future manage to find a new hack to display the battery info, this isn't the route I would take with my apps. I want to use officially supported APIs to make sure the apps keep working in the future, and I have no interest in risking my developer account by doing prohibited things. (Despite this approach, I have had to remove some features in the past, as Apple disallowed accessing certain APIs for security reasons.)