If there are no confining conditions whatsoever in respect of that item then yes to Pigeons and anything else you have permission to shoot. But most FACs will condition each weapon or group of weapons to a specific purpose. If your condition says Pest Control for that particular item then Pigeons are an acceptable quarry as are many other species.
However the condition may confine you further by specifying particular species or even circumstances. eg It may allow Pest control but only over certain land at certain times of the day (daylight hours).
It could, if the Police so wished, say you must stand on one leg and shoot over your shoulder with a mirror - I bet you wouldn't get very big bags then but at least in that instance you would have justification for appeal as it would in effect not allow you to use the gun.
