The most common and most likely answer to this crossword clue is the 5 letter word UMRAH
I've seen this crossword clue in The Guardian
U
M
R
A
H
Buddy explains!
The straight part
We know that "Pilgrimage" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case pilgrimage = umrah.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 UMRAH mean?
UMRAH noun- A minor hajj, a lesser pilgrimage to Mecca, other than at the time of the hajj.