Dia dhaoibh, a chairde

I've recently begun reading Mo Scéal Féin by Peadar Ua Laoghaire. In my readings have inspired questions that I find my current learning materials are insufficient to address. I've joined this forum in hope that the knowledgeable and enthusiastic members of this forum can help me figure some of these out.

I might as well just dive right in to the first one. In Chapter II there's the following sentence, which I've put in modern spelling Tá a fhios ag an saol nach féidir do leanbh, do gharsún nó do chailín bheag, sláinte cheart a bheith acu nuair a bhíd na fiacla ag leá ar an gcuma san amach as a gceann acu. I believe I've correctly parsed the meaning of the sentence as "Everyone knows it's not possible for a child, a boy or a little girl, to be healthy when their teeth are falling (melting) out of their heads like that."

My question is less about the meaning and more about the make of this sentence. It seems to me that sláinte is the real subject of the subclause. I therefore would've expected something like ...nach féidir do shláinte a bheith ag leanbh... instead of what's written. This expectation is based on analogy with sentences like ní féidir dom dul abhaile where it's the subject that comes after do. As far as I understand it, you wouldn't say ní féidir dom mé a dhul abhaile. So I guess in one sentence my question is this; why does the subject govern it's own verbal noun phrase, instead of coming after the preposition do?

Thank you in advance for reading this, I hope this isn't too much for a first question

(BTW: What do you mean by "subject"? The subject of the English or of the Irish sentence?
In Irish, what comes after do is part of the predicate (féidir do ...). The Irish (grammatical) subject is "sláinte a bheith acu" or "dul abhaile", the verbal noun phrase in total.)

You can't obviously say something like: "Is féidir dom í an rud a dhéanamh" ("It s possible for me that she makes the thing") because both parts have no logical connection. í is ungrammatical here.
You could only say: "Is féidir di an rud a dhéanamh" ("It s possible for her to make the thing")

But in PUL's sentence there is a logical connection and principally there is no need for verbal noun phrases (the Irish counterpart of English infinitive clauses) to have the same subject as the introducing phrase (as it is often the case in English).
And it is of course helpful that the logical subject of "sláinte a bheith acu" is "they" because it means "they have health"- so it is the same logical subject as in the "is féidir do" part.

That is so in every sentence of the kind "I can have it" (Is féidir liom é a bheith agam")
You wouldn't say: "Is féidir leis bheith agam" ("It can be at me" or even "It can be had by me"), either.

Good point this was fairly ambiguous. I meant to say what's usually the subject of the English clause comes after do. So in ní féidir dom dul abhaile I'd consider "I" the (logical) subject. My confusion stems from the fact that in this example sláinte is the subject of the verbal noun clause, but is distinct from the element I normally identify as the English subject, the noun/pronoun that comes after do. Since I'd never seen an example of this construction it caused me to wonder when exactly it's necessary or appropriate to specify the subject in the verbal noun phrase.

From your response I gather this is only necessary when the logical subject of the verbal noun phrase is both distinct from the grammatical subject and the same as the logical subject of the is féidir do... part of the sentence. So you don't say ní féidir dom mé a dhul abhaile because the subject of the verbal noun phrase is already apparent from dom. Similarly is féidir dom í an rud a dhéanamh is impossible because there's no logical connection between the subject of the verbal noun phrase and the is féidir do... phrase. In the event that Irish had a verb noun meaning "to have", say haváil, then it'd suffice to say ...nach féidir dóibh sláinte a haváil. It's only because Irish lacks such a verb and uses circumlocution with ag to show possession that we have to use Ua Laoghaire's construction. Is that right?

Yes.
But I don't want to establish complicated grammatical vs. logical subject rules. (They would lead astray.)
What I meant by "logical subject" is: The (logical and literal) meaning of a phrase is most important.

If the ability/possibility of someone has any impact on the verbal noun phrase contents, then the verbal noun phrase can be combined with an ability/possibilty phrase.
The meaning of "A bheith ag B" is "to have", so it can be combined with "is féidir do/le B".

Noteworthy is that Irish verbal noun phrases can have (grammatical) subjects on their own.
This is different to many other languages.

And bear in mind: Though "is féidir le/do" is used for English "can", it is very different syntactically.
English "I can X be at me" is strange and ungrammatical but "I have the ability that X is at me" is normal.
It is helpful to translate literally: is féidir liom/dom - "is possible* with me/for me"

(* féidir - Middle Irish étir - strength, so the etymologically correct literal translation is "is strength with/to me")

Where did the f come from to make étir into féidir?

Probably in analogy to the verb of similar meaning féad

A prefixed (or otherwise a dropped) f is very common in Irish (e.g. oscail / foscail, usa / fusa, agus / fogus, feiceáil / orig. ad-chiall, fanaim / org. anaim, fuacht / orig. uacht)
F is "dropped" in lenition (silent fh) and so often erroneously prefixed to words orginally start with a vowel.

As Labhrás was saying, the addition, and sometimes loss of an initial 'f' is a common freature of Irish.

The phenomenon is known as a prosthetic (prothetic)/ inorganic f. Its a form of hypercorrection. As Labhrá was saying, lenited f has no sound, so overtime, speakers became unsure whether a word etymologicall began with an f or a vowel; this led them to them to accidentally prefix f to some words which originally began with a vowel:

Old Irish : co n-accae ; Middle Irish: (f)acca
Old Irish: ás ; Middle Irish: fás etc...

This occurs even in the modern language as a dialectal marker:

Munster: faill ; Connacht/ Ulster: aill
Munster: fiolar; Connacht/ Ulster: iolar
Ulster: foscail; Connacht/ Munster: oscaill

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