The most common and most likely answer to this crossword clue is the 4 letter word RARE
I've seen this crossword clue in The Times
R
A
R
E
More clues leading to the result RARE
Buddy explains!

The straight part
We know that "Lightly-cooked" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case lightly-cooked = rare.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 RARE mean?
RARE noun- A scarce or uncommon item.
- Very uncommon; scarce.
- (of a gas) Thin; of low density.
- Good; enjoyable.
- (particularly meats) Cooked very lightly, so the meat is still red (in the case of steak or beef in the general sense).
- To rear, rise up, start backwards.
- To rear, bring up, raise.
- Early