The most common and most likely answer to this crossword clue is the 8 letter word NIBBLING
I've seen this crossword clue in The Daily Telegraph
N
I
B
B
L
I
N
G
More clues leading to the result NIBBLING
Buddy explains!

The straight part
We know that "having little taste" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case having little taste = nibbling.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 NIBBLING mean?
NIBBLING verb- To eat with small, quick bites.
- To bite lightly.
- To consume gradually.
- To find fault; to cavil.
- The act or sound of one who nibbles.
- A small adjustment or encroachment.