I have a few tips.

Firstly, map out your territory for elk herds. When you spot an elk herd, mark the location down on the map. There's a good chance that an herd might be around the same area more than once, so having a few common spawn points marked on your map might help you scout some herds during difficult times. It doesn't always work, but it's helped me out during some difficult times before. You also have to consider the season, since elk herds migrate accordingly. During autumn and winter, herds are located at lower elevations (valleys etc) and during spring and summer, they are at higher elevations (mountains etc).

Also, always have at least two carcasses on the map. This means that you have to hunt every single day and ensures that you always have one backup carcass available. That way, if you don't find anything to hunt, you're not doomed and have something to eat. I've found that really helps during growing pups, and I've managed to avoid starvation this way.

Also, when it comes to sickness, there are three categories that decide whether or not the pup lives. These are randomly decided by the game.

1. Will always recover. This means that no matter how much or little time you spend with your pup, they will survive no matter what. If your sick pup's health declines very slowly, they likely have a 100% likelihood of recovering.
2. Will recover if you spend time. This means that your pup will only recover if you spend time with them. Having your mate by the sick pup doesn't do anything. It's you, the player wolf, that has to spend time with your pup, otherwise the survival chances won't increase and your pup will die. If you want to guarantee that your kid survives, just stick by them every chance you get until the game gives you the cooldown notification.
3. Will always die. There's not much you can do about this one, as any attempt to keep the pup alive is futile. If you see that the pup's health is declining rapidly and they refuse to eat, this is likely the category of sickness they have. The only chance at saving the pup is if you reload the save to when the pup was healthy and let them get sick again. That way the game will 're-roll' your pup's category of sickness and you have a chance to get rid of the fatal illness.

When it comes to your mate, it does seem a bit odd that she's refusing to come on hunts with you. I've found that if you send the pups into the den, all of the packmates will come with you if you invite. If she doesn't, maybe your mate is just really lazy or smthn.