The most common and most likely answer to this crossword clue is the 7 letter word MEMPHIS
M
E
M
P
H
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S
More clues leading to the result MEMPHIS
- Everyman is taking speed in Elvis’ home
- Ruined city, one where a King was killed and another is buried
- Part of Tennessee Williams’ last “hip” production: morose, electrifying Marlon initially in the lead
- Maine politician belonging to ‘the man’s city’
- City speed is beyond me
- Motorway stress not as apparent in city
- Brussels ruled out of euphemism for birthplace of rock’n’roll
- Initially minimum stress, as leaving US city
- Everyman is taking speed in Elvis' home
Buddy explains!

The straight part
We know that "(the man’s, Elvis Presley’s) city" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case (the man’s, elvis presley’s) city = memphis.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.