You are in trouble with that pup if the cord is leaking as this is very likely infection. You can try cleaning it with peroxide or betadine but I'd suggest getting that pup to the vet for antibiotics and a close look at what is going on.
Make sure to keep bedding under the other pups clean and disinfected as if one pup has an infection its apt to spread.
Usually umbilical cords are dried up and gone by the end of the first few days.
Hope this helps you!
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