Which is more correct and preferable in accepting and rejecting the increase made by a trustworthy narrator: the methodology of the early Hadith scholars, who absolutely reject the solitary narration of a trustworthy narrator, or the methodology of the later scholars, who stipulate that the increase must be contradictory for it to be rejected? And what is the ruling on the addition of the phrase "he moves it (his finger), supplicates with it" in the Hadith of Wa'il ibn Hujr, based on that?

We have not seen the article in question. However, if a Hadith is narrated by several people, and one of them adds to their narration, scholars have differed on its acceptance. The majority of Hadith scholars and jurists are of the opinion that it should be accepted. If a narrator is unique in something that contradicts others, it is considered shādhdh (anomalous) and rejected. If there is no contradiction, then the narrator is considered. If they are trustworthy and possess a strong memory, their unique narration is accepted.

As for the Hadith of Wa’il ibn Hujr regarding moving the finger in the Tashahhud, it was narrated by Ahmad, An-Nasa’i, Abu Dawud, and others. It was authenticated by Ibn Khuzaymah, An-Nawawi, Al-Albani, and others. Those who declared it anomalous due to shudhudh overlooked the supporting narrations from trustworthy individuals. It was opposed by what was narrated from Az-Zubayr regarding the negation of movement, but the affirmative takes precedence over the negative, and the Hadith of Wa’il ibn Hujr is sounder.

Ibn ‘Uthaymeen (may Allah have mercy on him) said: The Sunnah indicates that one should clench the little finger and the ring finger, form a circle with the thumb and middle finger, raise the index finger, and move it during supplication.