Based on words like Q. pata-/S. pad- “walk” and S. pâd “step; track, road” (PE17/34), it seems Tolkien imagined a root ✱√PAT = “walk” or “step” in his later conception of the languages. This was probably a later iteration of unglossed ᴱ√PATA from the Qenya Lexicon of the 1910s with derivatives like ᴱQ. pata- “rap, tap (of feet)”, ᴱQ. patake “clatter”, and ᴱQ. patinka “shoe, slipper” (QL/72). Tolkien compared this early root to ᴱ√PETE which had derivatives like ᴱQ. pete- “knock, strike” (QL/73), so likely ᴱ√PATA¹ = “tap” (light) vs. ᴱ√PETE = “knock” (heavy). In the contemporaneous Gnomish Lexicon this root had derivatives like G. padra- “walk”, with a much clear connection Tolkien’s later ideas from the 1950s and 1960s. In The Etymologies of the 1930s, Tolkien gave N. pâd as an element Tharbad, whose gloss is unclear but seems to be “?Crossway” (Ety/THAR); this might indicate some continuity of thought between 1910s ᴱ√PATA¹ “✱tap” and 1950s/60s ✱√PAT “walk”.
The root ᴹ√PAT did appear in its own entry in The Etymologies of the 1930s, but it had derivatives like ᴹQ. panta-/N. panna- “to open” (Ety/PAT), making it more likely a continuation of ᴱ√PATA² or ᴱ√PṆTṆ “open, spread out, show” from the Qenya Lexicon of the 1910s, which coexisted with ᴱ√PATA¹ “✱tap” (QL/72). See the entry on ᴹ√PAT for further discussion.