I've been running the Amazon phone app on various tablets (hudl 2, memo pas HD and Tab4) for a while, and it all works fine - including prime streaming.. the only hard part is working out if the app you're installing is the phone app and not the tablet app - as it doesn't say phone in the name...
clearest instructions I found were:
To get prime instant video working you need to:
1. Enable settings/security/unknown sources
2. Download and install the Amazon Appstore using these instructions
http://www.amazon.co.uk/gp/mas/get/android/
3. Start the Appstore app, sign in, install the 'Amazon' and 'Prime instant video' apps (do not install the tablet version of the Amazon app - install the PHONE version)
4. Start the Amazon app, sign in and start watching.
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If the Appstore says that you have already purchased the app, tap on the appstore icon at top left, select 'My Apps' from the list, locate the app you need and click the install button.
PIV works fine on Sony Xperia Tablet Z and the new Tesco Hudl2. It works on Nexus 7 (old 2012 version) but with occasional stuttering.
from here:
http://www.amazon.co.uk/forum/kindle?_encoding=UTF8&cdForum=Fx3IRFCNF3E5K2W&cdThread=Tx17EWI5W7JF1CG
