The most common and most likely answer to this crossword clue is the 8 letter word DILUTION
I've seen this crossword clue in The Guardian
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More clues leading to the result DILUTION
Buddy explains!

The straight part
We know that "weakening" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case weakening = dilution.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 DILUTION mean?
DILUTION noun- The process of making something dilute.
- A solution that has had additional solvent, such as water, added to it into order to make it less concentrated.
- The process of bringing in unskilled workers to replace skilled ones, for example during wartime.
- Short for share dilution.