Counterpart to Confidante: Word for Someone Crying out for Help

"They .. sometimes seem to have little concept of the solace that a hurting human being can find in having someone else listen to and quietly acknowledge that person’s story, however awkward a listener might find it, without judgement, dogmata, moralizing or 'helpful advice' ('I was only trying to help!!!') that many use to soothe their own discomfort at the imperfect and awkward situation of the sufferer." (or person who is suffering)

Is this too obvious to be useful? I understand what you are looking for.

Difficult. I suggest suppliant (noun from adjective) as a possible alternative to supplicant although the meanings are almost identical.

Suppliant = humbly imploring : ENTREATING “a suppliant sinner seeking forgiveness”

Merriam Webster

A suppliant seeking advice. A suppliant seeking opinions or help.

“humble” has the specified connotation of vulnerability.

Etymologically, the word ought to be

• "miser" in the sense of "one who should be pitied" but that is archaic, or
• "wretch" in the sense of "one who is outcast",

But both those carry such heavy connotations that they would not be understood rightly in this sentence.

If it must be a single word, "sufferer" is probably your best bet, or perhaps "confider", but consider using an adjective, e.g. "unhappy soul"

To go by the title, the counterpoint to "confidante" is "confider" - the person who is doing the confiding.

I'm not sure in your example "their own discomfort at the imperfect and awkward situation of their confider." reads quite correctly but "of the person confiding in them" seems more accurate if a bit longer.