A root appearing in the Quenya Verbal System of the 1940s glossed “spit” with the derived verb ᴹQ. pis- of the same meaning (PE22/103). It may be a later iteration of the early primitive verb ᴱ✶pṣt- “spit”, first mentioned (unglossed) in The Qenya Phonology of the 1910s and again in Early Quenya Grammar of the 1920s as well as several Early Noldorin Word-lists from this same period (PE13/147, 163). Other earlier roots for spit include ᴱ√RETYE from Qenya Lexicon of the 1910s (QL/79), ᴱ√KEREKE from Gnomish Lexicon of the 1910s (GL/27), and ᴹ√PIW from The Etymologies of the 1930s (Ety/PIW).
Neo-Eldarin: For purposes of Neo-Eldarin, I would stick with the better-known root ᴹ√PIW from The Etymologies of the 1930s for verbs meaning “spit”, as it has derivatives in both Quenya and Noldorin. For nouns meaning “spit”, I personally adapt the early root ᴱ√KEREKE as a Neo-Eldarin root ᴺ√K(E)REK to salvage early Quenya and Gnomish words for spittle; see those entries for details.
A root in The Etymologies of the 1930s glossed “spit” with derivatives like ᴹQ. piuta- and N. puia- of the same meaning (Ety/PIW). See the entry ᴹ√PITH for a discussion of other roots of similar meaning.