The most common and most likely answer to this crossword clue is the 8 letter word DISPOSAL
I've seen this crossword clue in Independent
D
I
S
P
O
S
A
L
More clues leading to the result DISPOSAL
- Throwing away iPod, lass is upset
- Scrapping bribes contrarily pocketed by controller on set
- Set out to accept concessions brought back from management
- Loss paid off in assignment
- Clearance for use
- Management is very relaxed about bribes, on reflection?
- Work covered by girl I’d backed for management
- Face accommodating plea for help about power management
- Getting rid of old pass I shredded
- Phone about second point of sale and removal
Buddy explains!

The straight part
We know that "assignment" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case assignment = disposal.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 DISPOSAL mean?
DISPOSAL noun- An arrangement, categorization or classification of things.
- A disposing of or getting rid of something.
- The power to use something or someone.