The most common and most likely answer to this crossword clue is the 7 letter word UNGODLY
I've seen this crossword clue in The Daily Telegraph
U
N
G
O
D
L
Y
Buddy explains!

The straight part
We know that "Irreligious" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case irreligious = ungodly.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 UNGODLY mean?
UNGODLY adjective- Of a person: lacking reverence for God; of an action: not in accordance with God's will or religious teachings.
- Immoral, sinful, or wicked.
- Extreme; unreasonable.
- In an impious, irreverent, or ungodly manner; ungodlily.