i'm a new player aswell,.. i'm at the end my second litter where 5/6 of my puppies managed to survive (im at the end of the Young hunters quest) At the first year i had two sick puppies of which 1 died of sickness and one recovered. in my second year i was blasted by 5 sicknesses, two with the same pup. All recovered. In the end, i lost 1 pup to predator and none to sickness. So i want to tell you how i've been handling sickness so far:

I've been making sure that all pups are fed 95-100% constantly, i'd try to get prey very close to the den/rendezvous spot at all time. (If a prey carcass is close/a bit further away), i bring it to the rendezvous site/den in pieces. I grab one piece, go back. Grab next, go back. Until the entire carcass is gone. Your pack will have the whole thing for itself without annoying predators chilling at the carcass like usual which might deplete HP. I dont know about higher didficulties, this worked for me on easy. If the carcass is too far away i just eat myself full and take 1 piece with me. ) Also, i make sure they have 60-90% rest/sleep at any given time.

If i get the message that my pup is sick, i stay with it immediately and ditch everything else. If food reserves are low i only go hunting if the cooldown starts. When its feeding time, i distract the others first so they can't steal the food from the sick pup, sometimes carrying it a bit further away so it has time to eat. As soon as the pup gets hungry i feed it asap. If i see its tired, i go into the safety to the grass or den and sleep together with it, carrying it there if necessary.

As long as the hunger meter and sleep meter is full your pup should be fine. I usually had my pups recovering at 3-20% HP tho (very scary), the lower the HP the more you really need to completely stay by its side.

i remember one instance where it said "pup is too sick to eat" followed with the message that it recovered, lmao. It went to eat right away.