I think the holes are too deep for any sort of birds, but most of you seem to be saying Badgers create more mess, so I'm none the wiser. The holes are approx 3 inches deep, circular mostly. If it is Badgers, how can we deter them apart from stop feeding the birds (which we do but just fat balls and seeds in cages).
You could try leaving some unwashed clothes in the affected areas. The badger's sense of smell is second-to-none and they usually flee from human scent.
Clay soil - Cheshire/Derbyshire border. I play with plants and soil and sometimes it's successful
You can't deter badgers. You can keep them out with strong fencing, a metal mesh buried to about 9 - 12 inches into the ground. They seldom climb so it doesn't need to be very high.
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I play with plants and soil and sometimes it's successful