The most common and most likely answer to this crossword clue is the 7 letter word HARRIER
I've seen this crossword clue in The Sunday Telegraph
H
A
R
R
I
E
R
More clues leading to the result HARRIER
- Runner arrives in haste at start of relay
- Cross-country runner / who worries?
- One hounding prince, you say – and his mum?
- Dog in item of luggage, hot rather than cold
- Who would bother a runner?
- Runner who harasses somebody more like King Henry V!?
- Dog bird
- Avian canine
- Runner, girl, not finishing run
- Dog that flies!
Buddy explains!

The straight part
We know that "Dog/that flies!" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case dog/that flies! = harrier.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 HARRIER mean?
HARRIER noun- One who harries.
- Any of several birds of prey in the genus Circus of the subfamily Circinae which fly low over meadows and marshes and hunt small mammals or birds.
- A runner, specifically, a cross country runner.
- A kind of dog used to hunt hares; a harehound.