The most common and most likely answer to this crossword clue is the 4 letter word SELF
I've seen this crossword clue in The Times
S
E
L
F
More clues leading to the result SELF
- Troublemaker and others come down in street — sort of control needed
- Identity of second Santa helper
- Small boy upset about groups of Santa’s helpers being collected
- Most of human body turning seriously ill with internal issues?
- Volatile vet is fearless and forceful
- Will the author matter on reflection? Almost
- Single fleas in contortions forming their own bond
- Number one painting?
- Pompous worker chasing two mischievous children in class
- PM’s love need left out personal growth
Buddy explains!

The straight part
We know that "Pompous" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case pompous = self.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 SELF mean?
SELF noun- One individual's personality, character, demeanor, or disposition.
- The subject of one's own experience of phenomena: perception, emotions, thoughts.
- An individual person as the object of his own reflective consciousness (plural selves).
- Self-interest or personal advantage.
- Identity or personality.
- A seedling produced by self-pollination (plural selfs).
- A flower having its colour uniform as opposed to variegated.
- Any molecule, cell, or tissue of an organism's own (belonging to the self), as opposed to a foreign (nonself) molecule, cell, or tissue (for example, infective, allogenic, or xenogenic).
- To fertilise by the same individual; to self-fertilise or self-pollinate.
- To fertilise by the same strain; to inbreed.
- Having its own or a single nature or character throughout, as in colour, composition, etc., without addition or change; of the same kind; unmixed.
- Same, identical.
- Belonging to oneself; own.
- Of or relating to any molecule, cell, or tissue of an organism's own (belonging to the self), as opposed to a foreign (nonself) molecule, cell, or tissue (for example, infective, allogenic, or xenogenic).
- Himself, herself, itself, themselves; that specific (person mentioned).
- Myself.