This root did not explicitly appear until very late in Tolkien writing, in 1969-1970, but there were a few earlier precursors. Wynne, Smith, and Hostetter suggested the preposition Q. han “beyond” from Quenya prayers of the 1950s might be connected to this root. I think another precursor might be √ƷAN “long” or “extend” given as a variant of √YAN around 1959; Tolkien was considering √ƷAN as the basis for a new etymology of Q. anda/S. and “long” (PE17/40, 155), though there are no clear indications he carried through with this idea.
The root √ƷAN reappeared in 1968 notes with the gloss “adorn” serving as the basis for Q. antil, a word for the middle finger, perhaps being the place where rings were usually worn (VT47/26). As pointed out by Patrick Wynne this use of √ƷAN indicates it was from the period in the late 1960s where Tolkien was playing around with the form and phonetic developments of the ancient velar spirant [h] or [ɣ] (ʒ); see the discussion in the entry on how [[aq|initial [ɣ] became [h]]] in Ancient Quenya; this was one of the more common phonetic developments Tolkien used, but he vacillated a great deal on the subject.
√ƷAN reappeared again in 1969 notes on Quenya verbs with the gloss “give”, but here it was revised to √HAN (PE22/163). Tolkien then elaborated on the meaning of √HAN, saying that its actual sense was “enhance, enrich, add to”. In this note h- was preserved in Quenya and Telerin but lost in Sindarin, and the Quenya verb anta- “give” was influenced by but not directly derived from √HAN; in Sindarin the influence was more direct. We know that √HAN reappeared at least one more time in some notes from 1970 with the gloss “add to, increase, enhance, honour (espec. by gift)” (VT43/14) but this note remains unpublished so we don’t have the full context.
Sorting through all these variations is quite difficult, especially since they are intertwined with Tolkien’s shifting notions of the initial developments of ancient ʒ- and h-, but I think the basic conceptual development is (1959) √ƷAN “extend” >> (1968) √ƷAN “adorn” >> (1969) √ƷAN “give” >> (1969-70) √HAN “add to, increase, enhance, enrich, honour”.
Outside of its influence on “give” and “gift”, √HAN has no real attested derivatives, but it has long been theorized that it is basis for the Q. hantalë “thanksgiving” in Q. Eruhantalë (UT/214). This in turn serves as the basis for the most common Neo-Eldarin words for “thanks” and “to thank”, a pretty important word for polite conversation. These words, and their role in Neo-Eldarin, have been exceptionally contentious, giving this otherwise obscure root an outsized role in debates about Tolkien’s languages.