The most common and most likely answer to this crossword clue is the 6 letter word REOPEN
I've seen this crossword clue in The Times
R
E
O
P
E
N
More clues leading to the result REOPEN
Buddy explains!

The straight part
We know that "Make a fresh start" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case make a fresh start = reopen.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 REOPEN mean?
REOPEN verb- To open (something) again.
- To open again.