The most common and most likely answer to this crossword clue is the 5 letter word ISSEI
I've seen this crossword clue in The Times
I
S
S
E
I
More clues leading to the result ISSEI
Buddy explains!

The straight part
We know that "Immigrant" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case immigrant = issei.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 ISSEI mean?
ISSEI noun- A member of the first generation of Japanese immigrants to North America, South America or Australia.