The general licence is more of a conditions for shooting birds it's not a licence that you apply for like fac. Look it up. But it states things like to shoot pigeons they must be causeing damage to crops. You can shoot any animal that is classed as a pest and that is legal airgun quarry with a air gun, obviously mr fox and long ears are out of the question so are pheasants, partridge and so on.
But main ones you can shoot are pigeons; crows, magpies, jays, collard dove, rabbit, rat.
I suggest you have a good read of it and even print a copy off for such reasons as some bizzy body phones pol