The most common and most likely answer to this crossword clue is the 7 letter word PERSIST
I've seen this crossword clue in The Guardian
P
E
R
S
I
S
T
More clues leading to the result PERSIST
- Priests desperately keep on trying
- Is salesman turning up on street last?
- Strange priests carry on
- Last sibling gets fresh clothing
- Carry On Priests? Unlikely
- Carry on forward to embrace relative
- Carry on being cheeky about South Island
- Last relative tackled by forward
- Dodgy priests remain
- Name-dropping spinster resolved to keep going
Buddy explains!

The straight part
We know that "Carry On" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case carry on = persist.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 PERSIST mean?
PERSIST verb- To go on stubbornly or resolutely.
- To repeat an utterance.
- To continue to exist.
- To continue to be; to remain.
- To cause to persist; make permanent.