The sign of pus means the wound is already infected. I've never had an infected wound in our chickens but I'd probably do what I do to my own wounds. I know you say you don't have a place for her inside, but an infection left untreated is dangerous, and so is being outside in the cold. Do you have a crate (dog cage) or pet carrier you can put her in until she is dried and safe?
You need to cleanse the wound with a peroxide-water solution. I know peroxide can be painful but it's usually what most people have in the house and it's a powerful package. I would take her inside to do this since it's cold and being wet in cold weather would be very dangerous, especially in her condition. Once all the grit is out of the wound and you can see it clearly, make sure all the pus is drained out. It might be gross and nasty and the chicken may or may not put up a fight, so have someone else in the room with you to help hold her or do the cleaning. Once the wound is clean, I would apply some kind of antibiotic cream if you have any. If not, neosporin will probably have to do, and it's the only kind I have at home as well.
Apply that INSIDE the wound to kill the bacteria. Cover the wound - if it goes over her wings, so be it. Getting more dirt inside will be more dangerous than not being able to flap around, and it's probably best for her to remain still as possible to further healing.
Check the wound once a day at the least, more if you're worried, and apply more antibiotic cream as needed if it gets inflamed. You can also apply blue-kote on top of the neosporin because it actually works great at sealing the wound against external elements that may get through the bandage.
Is it possible to get a picture of the injury?
This is not expert advice, I'm just going off what I think would work best.
Is she acting more alert now that her neck is better? Is she still shaking?
Are her eyes clear? Make sure she is eating and drinking, as she needs the sustenance to promote her immune system!
