This root was connected to hand words for all of Tolkien’s life. Its periodic shifts between √MAH and √MAƷ had more to do with Tolkien’s ongoing vacillation on the form of the ancient velar spirant (voiced vs. voiceless) in Primitive Elvish. The earliest appearance of this root was as ᴱ√MAHA “grasp” in the Qenya Lexicon of the 1910s, with derivatives like ᴱQ. má “hand”, ᴱQ. maqa “handy, skilled”, and ᴱQ. māra “mighty, power, doughty; (of things) good, useful” (QL/57). In the contemporaneous Gnomish Lexicon the root was maχā with derivatives G. mô “hand” and G. manc “grip, grasp, hold” (GL/55); G. mora “good” may have also been related (GL/57).
In The Etymologies of the 1930s it appeared as ᴹ√MAƷ “hand” along with a related root ᴹ√MAG “use, handle”, with ᴹQ. mára “useful, fit, good (of things)” derived from the latter (Ety/MAƷ). In Tolkien’s later writings, the root √MAG was further differentiated from √MAƷ, becoming the basis for “good, useful”; see the entry on √MAG for further details. As for ᴹ√MAƷ “hand”, it was also mentioned in the Outline of Phonetic Development from the 1940s (OP1: PE19/48) and Outline of Phonology from the early 1950s (OP2: PE19/100 note #154).
In revisions to OP2, Tolkien decided the primitive velar spirant was unvoiced, a weak χ [x] rather than voiced ʒ [ɣ] (PE19/69 note #3; 72-73 note #22; 74 note #33). From this point forward, Tolkien sometimes rendered this root as √MAH (PE21/70; VT39/11), but it continued to appear often as √MAƷ even into the late 1960s (VT47/18-19) reflecting ongoing vacillation on the nature of the primitive velar spirant. As for its meaning in later writings, in Definitive Linguistic Notes (DLN) from 1959 Tolkien said it meant “serve, be of use” as opposed to words for “well” and “good” derived from √MAY or √MAG (PE17/162), and in another section of the same notes it was defined as “handle, manage, control, wield” (PE17/163).
Neo-Eldarin: For purposes of Neo-Eldarin, I think the last meaning is best for √MAH or √MAƷ. I think words for “good” and “useful” are better assigned to √MAG, and “excellence” to √MAY, though Tolkien often intermingled the meanings and derivatives of these three roots.
A verb in The Etymologies of the 1930s appearing in its Noldorin-style infinitive form tortho “ to wield, control” under the root ᴹ√TUR “power, control, mastery, victory” (Ety/TUR). As such it seems to apply mainly to the wielding of power and control of people. Compare to (S.) maetha- “to handle” for wielding physical objects.