Edit 5: The Stork's Whereabouts

[fuzzyllama1]

Interpretation of first look of the title without reading the body: A poem about waiting for a baby to come. Perhaps it will be a newlywed couple on their first child.
Interpretation after reading the body: Perhaps it is about infertility. It is still about a couple perhaps, but it is a little darker in theme than I initially thought. The man is consoling the woman, after once again being unsuccessful in becoming pregnant.

The Stork's Whereabouts

Though that stork is down and broken 
with its silken swaddle 
of stardust in the ocean's breath - Stardust makes another appearance. I'm assuming Stardust is the unborn child. After all, life originated from stardust. I think that is what you are going for here and I quite like it. I don't like starting this line with "of" (perhaps "silken swaddle, stardust in the ocean's breath").
as a sodden glimmer - Sodden as in saturated? Does this mean the glimmer is faint and soaked over or full bodied and bright? My guess is the former, but I'm not entirely sure. 
and a released farewell-handkerchief, - This is too long to go without any punctuation for me. I'm finding it hard to read with any sort of flow.
won't there always be a bird 
to give us that joy already given? - This strophe to me is saying the narrator still is hopeful and trying. The stork did not make it this time, but there will be plenty more opportunity.
 
And if that cloth has seen the ocean spilling - Consider cutting "the" here.
from the cliff,  
in stormy rips of currents
that our precious stardust rode, 
may we please at least go to the shore 
where that poor bird labors on the stone 
in his soggy clumps of crimson tufts, - Beautiful imagery of a bleeding, wounded stork. Soggy, crimson, clumps all take my mind one place (a period or maybe even miscarriage).
and bring him home for us to patch him up? - The narrator is asking (perhaps desperately) if there is anything they can do to have their stardust survive/delivered to them.
 
If not, I understand. 
Just stay in bed- 
we will bring you
chocolate, hugs, and tissues. - Consoling the fretting mother. I like how you use the cliche here. It is a unique perspective, so makes the chocolates/tissues thing is quite fresh in my opinion.

I am getting a dark vibe from this poem. My interpretation is this poem is about miscarriage or losing a carried child. There is some indication that you are describing the grieving process (depression, denial, and bargaining are ones that I can specifically point out). It is a beautiful and sad piece of writing. Good work.