The most common and most likely answer to this crossword clue is the 5 letter word EXUDE
I've seen this crossword clue in The Guardian
E
X
U
D
E
More clues leading to the result EXUDE
- Secrete 24th letter you had read out
- Put out when end of schedule’s unknown due to fog
- Discharge emptied underground enters English river
- Core of Brexit department in EU rejected issue
- River full of a lot of mud and discharge
- Former lover solver had heard weep
- Sweat, being examined by university dean? Not half!
- Leak from former lover you heard? Desperate, showing no heart
- Old model, due for exhibit
- Old duke, surprisingly, not originally known to sweat
Buddy explains!

The straight part
We know that "sweat" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case sweat = exude.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 EXUDE mean?
EXUDE verb- To discharge through pores or incisions, as moisture or other liquid matter; to give out.
- To flow out through the pores.