The short answer is unfortunately that it is impossible for apps to stay running in the background to check for new items in the clipboard. See below for the long answer!
The best solution possible, is therefore to make it as easy as possible to open the app each time you want to add a new item, and the easiest way currently is the Notification Center widget.
The long answer - TL;DR
Unfortunately, due to the way that iOS works, it is impossible for developers to keep the app open in the background, and therefore impossible to capture any new items added to the system clipboard unless the app is brought into the foreground. In fact, since iOS9, it is also impossible for an app to access the system clipboard when it is in the background, so even if the app could be kept running, it still couldn't access new items.
Previously, the only solution that anybody has found to this was a clever (but cheeky) method of playing a silent audio track constantly so that their app never went to sleep - because audio playback apps are the only ones allowed to stay running in the background. This not only used up dramatically more battery, but also was quickly banned by Apple. But anyway, even that would no longer work since iOS9 because of the limiting of access to the clipboard to foreground apps.
We'll of course be keeping an eye on the situation and will improve things wherever possible.
Unfortunately, due to the way that iOS works, it is impossible for developers to keep the app open in the background, and therefore impossible to capture any new items added to the system clipboard unless the app is brought into the foreground. In fact, since iOS9, it is also impossible for an app to access the system clipboard when it is in the background, so even if the app could be kept running, it still couldn't access new items.
Previously, the only solution that anybody has found to this was a clever (but cheeky) method of playing a silent audio track constantly so that their app never went to sleep - because audio playback apps are the only ones allowed to stay running in the background. This not only used up dramatically more battery, but also was quickly banned by Apple. But anyway, even that would no longer work since iOS9 because of the limiting of access to the clipboard to foreground apps.
This means that there's no app currently on the App Store that can constantly monitor the clipboard and add items when they're copied without user input.