Question Author
One of the school Governors is a policeman and he says it is a criminal offence to shoot somebody's cat with a 12-bore (though he would like to do it himself!).
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We have done springy asphalt on top of bark-chippings and removed it with remains of cat-poos still on it.
High wire tennis fence?- good idea, but not attractive in a village, and agile cats will find it no problem.
Definitely 'the cats', not hedgehogs.
Thankyou all so much who have responded so far!