Kept going stale, mysteriously, at the trading centre (6)
I believe the answer is:
lasted
'kept going' is the definition.
(I've seen this before)
'stale mysteriously at the trading centre' is the wordplay.
'mysteriously' is an anagram indicator.
'at' means one lot of letters go next to another.
'centre' says to take the centre.
The centre of 'trading' is 'd'.
'stale' anagrammed gives 'laste'.
'laste'+'d'='LASTED'
(Other definitions for lasted that I've seen before include "Continued for some time, endured" , "Continued, being salted" , "Kept going" , "Persisted, endured" , "Endured, survived" .)