The most common and most likely answer to this crossword clue is the 14 plus 10 letter word TRANSCENDENTAL MEDITATION
I've seen this crossword clue in The Daily Telegraph
T
R
A
N
S
C
E
N
D
E
N
T
A
L
M
E
D
I
T
A
T
I
O
N
Buddy explains!
The straight part
We know that "spiritual exercise" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case spiritual exercise = transcendental meditation.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.