If you have a sick , injured or orphaned pigeon please do not feed it immediately, it may be suffering from shock and/or dehydration.
It is important to give it a brief examination to establish any first aid that may be required.
When examining a pigeon I find it a lot easier on me and the bird (specially if it is a wood pigeon or a collared dove) to wrap it lightly in an old tea towel, or pillow case, or small hand towel, shielding the head and examining one part of the body at a time (eg left wing on top and underneath, same for right wing, chest, neck, vent, base of tail (look for fly eggs) feet, mouth and throat .
Stem any bleeding by direct pressure with a clean cloth.
If it is bleeding from the beak or a claw you can dip the beak or claw in cornflour to help stop the bleeding.
You should also check the inside of its mouth which should be clean and pink. Cheesy growths in the mouth could indicate canker.
Check its vent (under the tail). Sometimes a sick pigeon will get poop hardened over its vent, this will need washing off gently with a soft cloth or kitchen towel and warm water with a bit of salt added.
In hot weather it is particularly important to check it for . . The unhatched maggots will look like grains of white rice and will be found in the area of any wound, orifices such as the vent and sometimes at the base of feathers.
Check the front of its breast for damage which could indicate a ruptured crop.This is common in pigeons, it is repairable but will need a vet to suture both layers of the crop .
Check under its wings for wounds which could indicate it had been Cats carry the pasteurella bacteria in their saliva and a pigeon can die of pasteurella septicemia within 24 hours, so it is important that it is treated with antibiotics (preferably one that combines amoxicillin with clavulanic acid such as Synulox, Clavamox , Kesium or Clavaseptin) as early as possible.
If it feels cold it is important to warm it up on a hot water bottle wrapped in a towel (so it is warm but not hot) or on a heat pad set low, or under a 40 watt angled lamp for about an hour (keep an eye on it to ensure it doesn't overheat). If the pigeon is wet and hypothermic you it is best to use a hair dryer to dry and warm it but make certain that the air reaching the pigeon is not too hot. (Please note that if a bird is suffering from concussion heat could be harmful)