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Hmm, maybe. You can't access it directly, but maybe you can use a backup tool, which will then put a backup of the app data in an accessible spot. Whether you can do anything with that backup once you have it is a different question.
So, some options:
Helium is a backup app from the Play Store which does not require root. Whether you can install and run it with the current state of your phone's screen I'm not sure. You can install it by visiting play.google.com on a computer and selecting to install from there - that will push the app to your phone. But running it will require interaction with the phone.
ADB itself has an "adb backup" command, which if it works properly on your device should back up your data to your computer. I have read that it's not reliable on all models.
If you were rooted there would be other options, but then if you were rooted you could access the file directly using adb. Unfortunately on many devices the first step in rooting, unlocking the bootloader, wipes the device, which would be self-defeating here.
If those don't work, I suggest asking over in the Android Lounge section of the forum. That way more people will see the question than over in this corner, which gives us more chance of someone having a bright idea.