IANAL but the general licences are for specific purposes (see below). I doubt shooting in a back garden is covered by any of them.
BTW I am not anti shooting.
WML GL40: general licence to kill or take certain species of wild birds to conserve endangered wild birds and flora or fauna. Species covered: carrion crow, jay, magpie, Canada goose, Egyptian goose, monk parakeet, ring-necked parakeet, sacred ibis, Indian house-crow.
WML GL41: general licence to kill or take certain species of wild birds to preserve public health or public safety. Species covered: jackdaw, feral pigeon, Canada goose, monk parakeet.
WML GL42: general licence to kill or take certain species of wild birds to prevent serious damage and prevent the spread of disease. Species covered: carrion crow, jackdaw, magpie, feral pigeon, rook, woodpigeon, Canada Goose, monk parakeet, ring-necked parakeet, Egyptian goose, Indian House crow.