The most common and most likely answer to this crossword clue is the 10 letter word PILGRIMAGE
I've seen this crossword clue in The Daily Telegraph
P
I
L
G
R
I
M
A
G
E
More clues leading to the result PILGRIMAGE
- Sack Liberal leaving for dire crusade
- Resolute, 20, going through fortune to make significant journey
- Trip from drug — mostly horrid time
- I left, determined to block messenger in mission
- Journey that sees magi gripe, worried about length?
- Rifle’s used heartlessly guarding formidable mission
- Medicine’s short — not a good time for hajj
- Journey‘s formidable, stopping heartless plunder
Buddy explains!

The straight part
We know that "mission" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case mission = pilgrimage.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 PILGRIMAGE mean?
PILGRIMAGE noun- A journey made to a sacred place, or a religious journey.
- (by extension) A visit to any site revered or associated with a meaningful event.
- To go on a pilgrimage.