The most common and most likely answer to this crossword clue is the 6 letter word CENSER
I've seen this crossword clue in The Times
C
E
N
S
E
R
More clues leading to the result CENSER
Buddy explains!

The straight part
We know that "that swings round in some churches" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case that swings round in some churches = censer.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 CENSER mean?
CENSER noun- An ornamental container for burning incense, especially during religious ceremonies.
- A person who censes, a person who perfumes with incense