The most common and most likely answer to this crossword clue is the 5 letter word AMOUR
I've seen this crossword clue in The Times
A
M
O
U
R
More clues leading to the result AMOUR
- Lover of loud noise sacrificing little volume
- American and British close relationship
- The morning belonging to us for an affair
- Almost grieve after a love affair
- Grieve no end after a secret affair
- Mail perhaps dismissing top Republican’s affair
- University lecturer’s back after a second affair
- Who's endlessly keen after start of affair?
- A second primitive affair
- The morning belongs to us, love
Buddy explains!

The straight part
We know that "secret affair" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case secret affair = amour.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 AMOUR mean?
AMOUR noun- Love, affection.
- Courtship; flirtation.
- A love affair.
- A lover.