The most common and most likely answer to this crossword clue is the 4 letter word TANG
I've seen this crossword clue in The Daily Telegraph
T
A
N
G
More clues leading to the result TANG
Buddy explains!

The straight part
We know that "tastefulness" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case tastefulness = tang.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 TANG mean?
TANG noun- A refreshingly sharp aroma or flavor.
- A strong or offensive taste; especially, a taste of something extraneous to the thing itself.
- A sharp, specific flavor or tinge.
- A projecting part of an object by means of which it is secured to a handle, or to some other part.
- The part of a knife, fork, file, or other small instrument, which is inserted into the handle.
- The projecting part of the breech of a musket barrel, by which the barrel is secured to the stock.
- The part of a sword blade to which the handle is fastened.
- A group of saltwater fish from the Acanthuridae family, especially the Zebrasoma genus.
- (games) A shuffleboard paddle.
- Tongue
- (by extension) Anything resembling a tongue in form or position such as the tongue of a buckle.
- A sharp, twanging sound; an unpleasant tone; a twang.
- To strike two metal objects together loudly in order to persuade a swarm of honeybees to land so it may be captured by the beekeeper.
- To make a ringing sound; to ring.
- Knotted wrack, Ascophyllum nodosum (coarse blackish seaweed)
- The vagina.
- Intercourse with a woman