Why isn't my dog's red, oozy neck sore healing with home care?

Unfortunately hydrogen peroxide doesn't help pets with any type of skin condition. In fact we don't use it for any wounds because it can potentially cause cell necrosis (dying off of normal tissue). This appears to be a deep infection, which is likely why it hasn't healed well for you.
You can try cleaning it with Betadine diluted 50:50 with warm water and the hair needs to be shaved around this to allow the skin to breathe and heal well. I would not place Neosporin on a wound like this, because it likely won't help. You may be able to find a hot spot spray at your local pet store that can be used.
However, there is a good chance this won't heal without the help of your vet, who has stronger oral antibiotics and topical medical sprays to take care of this properly. I would schedule an appt if the Betadine and clip are not able to improve this area well within the next few days. Best wishes getting Murphy's skin to heal. Take care.