My nursing dog's teats are sore from puppy nails. What can I do?

I am very sorry to hear that Lucy is having these problems while she is nursing her pups. It sounds like you are very on top of the situation and attentive to her needs. Since the pups are four weeks old hey will have started to wean, which I can see you have done. This means that Lucy can have a break too. I would place a tshirt on Lucy and let her tears have a break for a couple hours each day.

Since she was also warm I would watch for signs of mastitis. Mastitis is a bacterial infection of one, or sometimes more then one, of the milk producing glands in the breasts. Other clinical signs would be;

Swelling
Firm
Warm
Discharge (other then milk)
painful to the touch
Loss of appetite
Lethargy
Neglect of puppies

If you suspect that Lucy could have mastitis based on the above clinical signs I would suggest taking her to see a veterinarian. The vet will be able to prescribe an antibiotic, antiinflamatory medications and pain relief.

In the meantime you can apply a cool compress to the affected area. Do not attempt any over the counter medications or ointments as they may cause pain or delay healing.

I hope this was helpful and more importantly that Lucy starts to feel better soon.