The most common and most likely answer to this crossword clue is the 6 letter word FUSSED
I've seen this crossword clue in The Guardian
F
U
S
S
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 = fussed.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 FUSSED mean?
FUSSED verb- To be very worried or excited about something, often too much.
- To fiddle; fidget; wiggle, or adjust
- (especially of babies) To cry or be ill-humoured.
- (with over) To show affection for, especially animals.
- To pet.
- (usually in the negative) Bothered; concerned.