The most common and most likely answer to this crossword clue is the 3 plus 5 letter word ART HOUSE
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Buddy explains!

The straight part
We know that "independent cinema" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case independent cinema = art house.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 ART HOUSE mean?
ART HOUSE noun- A cinema that shows art films and foreign films which are not widely distributed.
- A building or gallery in which works of art are collected, displayed, and offered for sale.