Here in the printer’s manuscript, Oliver Cowdery initially wrote “sons of a liar”; then he crossed out the word sons and inserted children supralinearly (there is no change in the level of ink flow). This correction does not seem to be due to editing; instead, it appears to be a correction to the original manuscript (no longer extant here). A few verses later on, in Alma 20:13, we have a case of “sons of a liar”, but verse 13 also uses children to refer to these “sons of a liar”:
So the use of “the children of a liar” in Alma 20:10 is possible. There are no other instances in the text of either “sons of a liar” or “children of a liar”, but since either sons or children can theoretically be used in this expression, the critical text will follow Oliver Cowdery’s virtually immediate correction in verse 10 to “children of a liar”.
Summary: Accept in Alma 20:10 Oliver Cowdery’s almost immediate correction in 𝓟 of “sons of a liar” to “children of a liar”; either reading is possible, which means that the corrected reading in 𝓟 was probably the reading in 𝓞.