The most common and most likely answer to this crossword clue is the 7 letter word REFUGEE
I've seen this crossword clue in The Times
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More clues leading to the result REFUGEE
- Asylum seeker to cater for again, mostly, embracing universal good
- Exile concerned with stale atmosphere vacated enclosure
- Being in flight, not agree say to turn back for son
- Judge associated with the top people: I'm surprised he's been driven out
- One escaping English stuffiness cuts free, going topless
- Big leader off, free to move around as outcast
- Neanderthal comment by judge on Europeans, one seeking shelter
- Fugitive from EU free to wander round heart of Belgium
- United following umpire -- gosh, he's fleeing
- Whistle-blower with no right to encourage Eastern asylum seeker
Buddy explains!

The straight part
We know that "Asylum seeker" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case asylum seeker = refugee.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 REFUGEE mean?
REFUGEE noun- A person seeking refuge in a foreign country out of fear of political persecution or the prospect of such persecution in their home country, i.e., a person seeking political asylum.
- A person seeking refuge due to a natural disaster, war, etc.
- A person formally granted political or economic asylum by a country other than their home country.
- (by extension) A person who flees one place or institution for another.
- To convey (slaves) away from the advance of the federal forces.