The most common and most likely answer to this crossword clue is the 6 letter word FANDOM
I've seen this crossword clue in The Times
F
A
N
D
O
M
Buddy explains!

The straight part
We know that "Supporters generally" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case supporters generally = fandom.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 FANDOM mean?
FANDOM noun- The fans of a sport, activity, work, person etc., taken as a group.
- The subculture of fans.
- The state, quality, or condition of being a fan.