Solved!

I'd thought that it was the hoards* of greenfinches, but there may be another solution to the rapidly disappearing sunflower hearts....

We've been hearing the Great Spotted Woodpecker a fair bit recently, and Cathy said that she'd seen it in the garden.  Finally, I got to see it too, although I had to put the phone down mid-conversation so as to be able to  run and get my camera when the bird arrived on the feeder for this shot.  Sure enough though, after a few minutes the greenfinches arrived and discouraged the woodpecker.

*Well, not hoards exactly, but they are now stopping by in groups of up to 8.  Technically I believe that the collective noun for finches is a charm.  Love them though I do, I'm not sure that its terribly apposite for greenfinches.