I had a similar problem to this! Except mine occurred when en route to a crucible match; music went all glitchy, picture got really framey, etc. Then when I rebooted the game tried to update, it got stuck at 16% verifying content.
With no better ideas, I followed advice from an xbox-related thread (cringe) and tried putting a different game in, loading that up, quitting it and trying destiny again. Lo and behold, it worked! Fingers crossed that it doesn't do it again...
Hope this helps! :)
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Thanks very much for your help Mikkyday :) I just started the game to try your suggestion and it loaded normally right away. I hadn't played another game since, but I did watch 4OD, then turned it off over night. I don't know if loading 4OD helped? I had turned turning it off quite a few times before that. Thanks again. It was very frustrating last night, pleased to have to working again :)