The most common and most likely answer to this crossword clue is the 8 letter word DOGGEDLY
I've seen this crossword clue in The Sunday Telegraph
D
O
G
G
E
D
L
Y
More clues leading to the result DOGGEDLY
Buddy explains!

The straight part
We know that "with determination" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case with determination = doggedly.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 DOGGEDLY mean?
DOGGEDLY adverb- In a way that is stubbornly persistent.
- Sullenly, gloomily