The most common and most likely answer to this crossword clue is the 4 letter word GYMS
G
Y
M
S
More clues leading to the result GYMS
Buddy explains!

The straight part
We know that "Generally members have their middles suppressed here" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case generally members have their middles suppressed here = gyms.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 GYMS mean?
GYMS noun- A sports facility specialized for lifting weights and exercise.
- Physical education class
- To go to the gym.
- A large room or building for indoor sports.
- A type of secondary school in some European countries which typically prepares students for university.
- A public place or building where Ancient Greek youths took exercise, with running and wrestling grounds, baths, and halls for conversation.
- A sport involving the performance of sequences of movements requiring physical strength, flexibility, and kinesthetic awareness.
- Complex intellectual or artistic exercises or feats of physical agility.