she may have ruptured a little blood vessel in her vent as she attempted to lay an egg or passed a particularly hard dropping. This happens sometimes and is usually just a one off event but if you notice that she is continuing to pass blood or if she develops a protuberance around the vent area, you may need to take her to see a vet.
If there is anything sticking out of the vent area this could be a prolapse and this is where the oviduct or egg laying tube has been pushed out of the body in an effort to lay an egg. It is imperative that you separate her from other hens as they will be attracted to the redness and peck at it which can cause your hen distress and serious injuries
If there is obviously anything unusual hanging out of the vent, wash the prolapsed area with warm water with some antiseptic in it. I cannot see her vent clear enough in your post to tell if there is or not.
If she has ruptured a little blood vessel in her vent because she's struggling to lay a large egg, you may be able to help her to pass it by warming up the vent area either by sitting the hen in a bowl of warm water or on a towel covered hot water bottle or by gently blowing warm (not hot) air from a hairdryer onto the area. This, accompanied by a little Vaseline or olive oil rubbed onto the vent is usually enough to relax the muscles and allow the egg to be passed. It’s probably best to bring her in to the house so that she’s nice and warm and you may find that she lays an egg and returns back to normal soon afterwards. If you give her a towel covered hot water bottle to lie on and keep her calm in a darkened room. An egg bound hen will look miserable, hunched and fluffed up. She may appear to be straining to lay an egg and she may also be panting and breathing fast. If you feel her abdomen and vent area, you might be able to feel the egg.
There's a chance that bleeding has occurred because your hen has been pecked in the vent area by another hen as this is a prime target for bullies. If you find that this is the case, you can get a variety of sprays to use on the bullied bird which make her feathers taste unpleasant to the other birds and these are usually enough to deter pecking. When the skin is red, sore or broken, it is important to separate the injured hen straight away and you can use Veterinary Wound Powder on them to help stem the bleeding and promote healing. Hens are morbidly attracted to the colour red and will peck at wounds until they are in a dreadful state if nothing is done so Gentian or Purple Spray is very effective as it stains the skin purple and this makes it a much less obvious target for the bully. If a hen develops a wound of any kind, remove them and allow it to heal for a few days before reintroducing them to prevent the wound being pecked.
