Crossword Genius

Add to men relieving us in August (7)

Ross

I believe the answer is:

augment

I'm a little stuck... Click here to teach me more about this clue!

'add to' is the definition.
(I've seen this before)

'men relieving us in august' is the wordplay.
I cannot really understand how this works, but
'us' could be 'me' ('us' can mean 'mean' in some UK dialects) and 'me' is found in the answer.
'august' could be 'aug' (short for August) and 'aug' is located in the answer.
This explanation may well be incorrect...

Can you help me to learn more?

(Other definitions for augment that I've seen before include "Swell" , "Make larger" , "Enlarge or increase" , "Expand" , "Make greater in size, number, etc." .)

I've seen this clue in The Sun.
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