What is the difference between a shadh (irregular) hadith and an addition by a reliable narrator, and how does this apply to the Wāʾil ibn Ḥujr hadith regarding moving the finger in the tashahhud?

The relationship between a shadhdh (anomalous) hadith and the addition by a reliable narrator is that if the addition comes from someone whose singular narration is not accepted, then it is considered shadhdh and rejected.

Ibn al-Salah elaborated that if a narrator singles out a narration that contradicts what was narrated by someone more reliable than him, then what he singly narrated is shadhdh and rejected. However, if there is no contradiction, then one looks at the solitary narrator: if he is a reliable and accurate preserver (of hadith), then what he singly narrated is accepted; otherwise, his singular narration would be a defect in it.

Regarding the hadith of Hujr ibn Wa'il: Zaidah ibn Qudamah was the only one who narrated the addition "he moves it." Al-Albani authenticated this addition and deemed it not shadhdh, mentioning that Ibn Khuzaymah, Ibn Hibban, and others had authenticated it.

In conclusion, the hadith of Wa'il with the addition of the movement is authentic, and it has corroborating narrations in its meaning. Those who deemed it shadhdh overlooked the narrations of reliable narrators that concur with it and the authentication by the early Imams.