The most common and most likely answer to this crossword clue is the 10 letter word DOWNSTREAM
I've seen this crossword clue in The Times
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More clues leading to the result DOWNSTREAM
- With current having failed, right to probe old energy source?
- Fantasise about private thoroughfare where the current flows
- With the current duke, have new master
- In direction of current ambition, securing private street
- Quickly finish drink by moving arm, set towards mouth
- Eat seconds with squad, carrying last of dinner towards the mouth?
- The current trend, naturally
- Move towards men, going with the flow
Buddy explains!

The straight part
We know that "With current" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case with current = downstream.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 DOWNSTREAM mean?
DOWNSTREAM verb- To stream downward.
- Lower down, in relation to a river, stream or flow of fluid
- In the direction from the server to the client.
- Towards the 3' end of a DNA molecule.
- (metaphoric) Influenced by something; being a consequence of something.
- Following the path of a river or stream