The most common and most likely answer to this crossword clue is the 7 letter word REEKING
I've seen this crossword clue in Independent
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More clues leading to the result REEKING
Buddy explains!

The straight part
We know that "Very smelly soldiers beginning to exasperate His Maj? (7)/Very smelly" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case very smelly soldiers beginning to exasperate his maj? (7)/very smelly = reeking.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 REEKING mean?
REEKING verb- To have or give off a strong, unpleasant smell.
- To be evidently associated with something unpleasant.
- To be emitted or exhaled, emanate, as of vapour or perfume.
- To emit smoke or vapour; to steam.
- Totally stinking.