The most common and most likely answer to this crossword clue is the 6 letter word NUMBER
I've seen this crossword clue in The Times
N
U
M
B
E
R
More clues leading to the result NUMBER
Buddy explains!

The straight part
We know that "One possibly/less sensitive" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case one possibly/less sensitive = number.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 NUMBER mean?
NUMBER noun- An abstract entity used to describe quantity.
- A numeral: a symbol for a non-negative integer.
- An element of one of several sets: natural numbers, integers, rational numbers, real numbers, complex numbers, and sometimes extensions such as hypercomplex numbers, etc.
- (Followed by a numeral; used attributively) Indicating the position of something in a list or sequence. Abbreviations: No or No., no or no. (in each case, sometimes written with a superscript "o", like Nº or №). The symbol "#" is also used in this manner.
- Quantity.
- A sequence of digits and letters used to register people, automobiles, and various other items.
- A telephone number.
- (grammar) Of a word or phrase, the state of being singular, dual or plural, shown by inflection.
- (in the plural) Poetic metres; verses, rhymes.
- A performance; especially, a single song or song and dance routine within a larger show.
- A person.
- An item of clothing, particularly a stylish one.
- A marijuana cigarette, or joint; also, a quantity of marijuana bought form a dealer.
- An issue of a periodical publication.
- A large amount, in contrast to a smaller amount; numerical preponderance.
- To label (items) with numbers; to assign numbers to (items).
- To total or count; to amount to.
- Physically unable to feel, not having the power of sensation.
- Emotionally unable to feel or respond in a normal way.
- Causing numbness.