Aquilegia,
I know about your problem so well because we suffer from the same one.
years ago, I had a bird table which I surrounded with chicken wire (with holes about 5cm square). This would allow in those smaller birds - i.e. robins, chaffinges, tits, and leave out the larger ones.
However, it will not allow in the blackbirds and fieldfairs. So you will need to leave some food for them on the grass. Blackbirds and fieldfairs are very early feeders and will get their full before the dreaded pigeons get there!
Happy birding ! :)