Xcode has a built-in feature for inspecting the filesystem of developer installed apps on an iOS device.

To do this, connect your device to your Mac and select Window > Devices in the Xcode menubar. Then select your device in the left pane under the Devices section. This will show a list of developer installed apps for that device. Select the app you want to inspect and then the select the gear icon near the bottom of the screen. Here you can view the current filesystem by selecting Show Container or download a snapshot of it. Selecting Download Container... will download and export a snapshot of the filesystem as a .xcappdata file that you can browse through. Right-click on this file and select Show Package Contents to open the folder. Open the AppData folder and you should now see a few folders like Documents, Library, tmp, etc.

The AppData/Library/Caches/com.ORGANIZATION.APPNAME folder may have a Cache.db file that contains the shared url cache for the app. So, if you needed to view the cached response urls, you can execute sql queries on that database file. For example, in the Terminal, cd to the AppData/Library/Caches/com.ORGANIZATION.APPNAME directory and fire up sqlite by entering sqlite3 Cache.db. Entering .tables will give you a list of all tables for that database. Here you should see the cfurl_cache_response table. Running SELECT * FROM cfurl_cache_response will then print a list of all app-wide url responses that you can inspect.