The most common and most likely answer to this crossword clue is the 4 letter word NUMB
I've seen this crossword clue in The Guardian
N
U
M
B
More clues leading to the result NUMB
- Maybe one has no hesitation becoming unfeeling
- Senseless maiden in retro hairstyle
- Dead queen ignored by count
- Dead centre, miles inside
- Unfeeling Greek character, one with medical qualification
- Stupefied, having taken umbrage a little
- Blunt military leader in essence
- No expression of doubt in song having no feelings
- “Dead meat” head of Mafia admitted
- Insensitive one, for example, displays no hesitation
Buddy explains!

The straight part
We know that "Insensitive" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case insensitive = numb.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 NUMB mean?
NUMB verb- To cause to become numb (physically or emotionally).
- To cause (a feeling) to be less intense.
- To cause (the mind, faculties, etc.) to be less acute.
- To become numb (especially physically).
- Physically unable to feel, not having the power of sensation.
- Emotionally unable to feel or respond in a normal way.
- Causing numbness.