There is a way to enable additional push debug messages:
Enabling Push Status Messages on iOS
To enable logging on iOS, install the configuration profile PersistentConnectionLogging.mobileconfig on your device by either putting the file on a web server and downloading it using Safari on your device, or by sending it as an email attachment and opening the attachment in Mail on your device. You'll be prompted to install "APS/PC Logging".
Apple Developer TechNote