Hi Mark!
So I raise Muscovy ducks, so all my info is based around them and what I've experienced raising them.
My ducklings that I hatched this year were not lethargic after hatching - after a few hours of coming out of the eggs, they were running all over the place! I have another post on here about some trials with that, I'll link it - you may want to try separating your broody hens somehow, since I ran into issues with one batch hatching before the others did. The chicks seemed to trigger mothering instincts, as well as unfortunately having one baby get crushed in a different nest of eggs. It was a real mess.
With the piece of egg attached, that sounds kinda normal to me, maybe someone else can chip in? I would guess a mother will help a baby clean up, but it sounds like this one is unclaimed so might not have gotten the help it needed to get the sticky stuff off. There's all kinds of blood and mucous in the egg that is shed when they hatch.
When researching ducks, I read that muscovies are considered really good moms (and aside from the issues right at hatching, they've been great) and that some other breeds are not. You should look into that, because I feel like I remember seeing that pekings are not good mothers. You might be better off getting an incubator and hatching there, or using a good mother to hatch your babies.
This is purely speculation, but the way I look at it is that simple minded animals like this have some triggers that need to be tripped for them to go into different modes of care - you see it when they go broody, and I felt like they switched again when the ducks had hatched. It might be that one baby hatching wasn't
enough to trigger that, and they're considering it an anomaly. One duckling hatching and needing different care isn't enough to abandon a nest full of eggs to take care of it.
Like I said, mine were up and running pretty quickly - hatched mostly at night, and running around underfoot by the next morning.
You might have some luck taking care of it inside, and once a big batch of babies has hatched, slip it back into the mix. In my other post I talked about that; one duckling had been abandoned (due to malnutrition I believe) and I nursed it back to being able to walk, and gave it back to the mother within a few hours. There were no issues for it being accepted back in, but you might see some if it's been a few days. You might just have one duck in your bathtub for a month. :)
To nurse my little malnutritioned guy back, I was doing the same as you, get some water in it's bill, then mix food and water into a mush, and get it eating that. After it's up and running around, raise it like normal! I set up a lamp in the bathtub with
straw, and a food and water bowl, and raised my starter flock there for almost a month before I got sick of them and stuck them on the deck lol.
I see it's been a couple days since you posted, I hope everything is all right!
https://permies.com/t/161939/Bad-Experience-Broody-Muscovies
And here's my post about the trainwreck we had a month ago. :(