The most common and most likely answer to this crossword clue is the 4 letter word LIED
I've seen this crossword clue in The Times
L
I
E
D
More clues leading to the result LIED
Buddy explains!

The straight part
We know that "Was lacking in veracity, primarily" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case was lacking in veracity, primarily = lied.The cryptic/wordplay part
In cryptic crossword there's often an wordplay/cryptic part as well.This part involves a more complex hint, such as an anagram, homophone, hidden word, etc.
Sadly, We don't have an explanation for this specific crossword clue yet.
What does LIED mean?
LIED noun- An art song, sung in German and accompanied on the piano.
- To give false information intentionally with intent to deceive.
- To convey a false image or impression.
- To be mistaken or unintentionally spread false information.