The most common and most likely answer to this crossword clue is the 4 letter word DUEL
I've seen this crossword clue in The Daily Telegraph
D
U
E
L
More clues leading to the result DUEL
Buddy explains!

The straight part
We know that "head-to-head fight" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case head-to-head fight = duel.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 DUEL mean?
DUEL noun- Arranged, regular combat between two private persons, often over a matter of honor.
- Historically, the wager of battle (judicial combat)
- Any struggle between two contending persons, groups or ideas.
- To engage in a battle.