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**Python 654 bytes**

```python
L,H,E=range(3);V=set("eéiíaáoóuúEÉIÍAÁOÓUÚ");r={"mé":("mo",L),"tú":("do",L),"sí":("a",H),"sé":("a",L),"muid":("ár",E),"sibh":("bhur",E),"siad":("a",E)};s=lambda a,b:a.upper() if b[0].isupper() else a
def f(x,y):w,rule=r[x];w=s(w[0]+"'" if w in{"mo","do"}and y[0]in V else w+" ", y);return w+[l,lambda v:"h"+v if v[0] in V else v,e][rule](y)
def l(w):
 if w[:2].lower() in "sc sf sm sp st".split():return w
 if w[0].lower() in set("bcdfgmpst"):return w[0]+s("h",w)+w[1:]
 return w
def e(w):
 for r in"bm cg dn fbh gn pb td".split():
  if w[0].lower() == r[0]:return r[1:]+w
 if w[0] in V:return"n"+w if w[0].isupper() else "n-"+w
 return w
```

the original, longer version is

```python
vowels = set("eéiíaáoóuúEÉIÍAÁOÓUÚ")

def f(x, y):
    possessive_pronoun, rule = rule_dict[x]
    if possessive_pronoun in {"mo", "do"} and y[0] in vowels:
        possessive_pronoun =  same_case(possessive_pronoun[0], y) + "'"
    else:
        possessive_pronoun = possessive_pronoun + " "
    if y[0].isupper():
        possessive_pronoun = possessive_pronoun.upper()

    if rule == "lenition":
        y = lenition(y)
    if rule == "h-prothesis":
        y = h_prothesis(y)
    if rule == "eclipsis":
        y = eclipsis(y)
    return possessive_pronoun + y

def h_prothesis(word: str) -> str:
    if word[0] in vowels:
        return "h" + word
    return word

def lenition(word: str) -> str:
    l1 = "sc sf sm sp st".split()
    if word[:2].lower() in l1:
        return word
    if word[0].lower() in set("bcdfgmpst"):
        return word[0] + same_case("h", word) + word[1:]
    return word

def same_case(a, b) -> str:
    if b[0].isupper():
        return a.upper()
    return a.lower()

def eclipsis(word: str) -> str:
    rules = ["bm", "cg", "dn", "fbh", "gn", "pb", "td"]
    for r in rules:
        if word[0].lower() == r[0]:
            word = r[1:] + word
            return word
    if word[0] in vowels:
        if word[0] != word[0].lower():
            return "n" + word
        else:
            return "n-" + word
    return word

rule_dict = {
    "mé":   ("mo", "lenition"),
    "tú":   ("do", "lenition"),
    "sí":   ("a", "h-prothesis"),
    "sé":   ("a", "lenition"),
    "muid":	("ár", "eclipsis"),
    "sibh":	("bhur", "eclipsis"),
    "siad":	("a", "eclipsis"),
}
```

You can test with this code:

```python
from test_cases import cases

from irish_possessive_golf import f

def test_all():
    for i, (x, y, result) in enumerate(cases):
        my_result = f(x, y)
        assert my_result == result, f"{x}, {y} should give {result}, not {my_result} ({i+1})"
    print("All succeeded")

test_all()
```