The most common and most likely answer to this crossword clue is the 10 letter word MAINTAINED
I've seen this crossword clue in The Daily Telegraph
M
A
I
N
T
A
I
N
E
D
Buddy explains!

The straight part
We know that "carried on" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case carried on = maintained.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 MAINTAINED mean?
MAINTAINED verb- To support (someone), to back up or assist (someone) in an action.
- To keep up; to preserve; to uphold (a state, condition etc.).
- To declare or affirm (a clause) to be true; to assert.
- Showing maintenance or attention