The most common and most likely answer to this crossword clue is the 4 plus 2 letter word KEEP AT
K
E
E
P
A
T
Buddy explains!
The straight part
We know that "Persist with" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case persist with = keep at.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.