We had the same trouble, and used plastic coated wire that covers halfway between the table and the roof, allowing little birds access, but keeps out pigeons - and its worked a treat. I agree they are a nuisance - like mini vacuum cleaners with the food.
poor hungry pigeons.. arrr. I put my 2 week old puppies in an open pen in the garden with their mother for some sunshine six of them. When I heard their mother barking I ran out to see; She had left the pen for a toilet moment; about 20 pigeons all lined up on the electric wires gazing greedily at the pups a couple of them had swooped at the pups which caused the barking. Shoot the varmints.
My grandparents used to have those wooden windmill things dotted about the garden - they seemed to discourage big birds but not the little ones - you could try with one of those plastic windmill on a stick things attached to your bird table.