The most common and most likely answer to this crossword clue is the 5 letter word BRASH
I've seen this crossword clue in The Daily Telegraph
B
R
A
S
H
More clues leading to the result BRASH
- Self-assertive bachelor lacking prudence
- Britain remains bold
- Supporters rush finally forward
- Head of bank, reckless and cocky
- Silence follows supporter who’s offensively bold
- Cocky supporter’s hard to sit behind
- Book reckless forward
- Cocky Republican infiltrating party
- Flashy bastard lacking caution
- Crude rock fragments
Buddy explains!

The straight part
We know that "tasteless and offensive" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case tasteless and offensive = brash.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 BRASH mean?
BRASH noun- A rash or eruption; a sudden or transient fit of sickness.
- A sudden burst of rain.
- An attack or assault.
- To disturb.
- (of people or behaviour) Overly bold or self-assertive to the point of being insensitive, tactless or impudent; shameless.
- (of actions) Overly bold, impetuous or rash.
- (of things) Bold, bright or showy, often in a tasteless way.
- Leaf litter of small leaves and little twigs as found under a hedge.
- Broken and angular rock fragments underlying alluvial deposits.
- Broken fragments of ice.
- Brittle (said e.g. of wood or vegetables).