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

The straight part
We know that "Went/some dope" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case went/some dope = weed.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 WEED mean?
WEED noun- Any plant regarded as unwanted at the place where, and at the time when it is growing.
- Short for duckweed.
- Underbrush; low shrubs.
- A drug or the like made from the leaves of a plant.
- A weak horse, which is therefore unfit to breed from.
- A puny person; one who has little physical strength.
- Something unprofitable or troublesome; anything useless.
- To remove unwanted vegetation from a cultivated area.
- A garment or piece of clothing.
- Clothing collectively; clothes, dress.
- An article of dress worn in token of grief; a mourning garment or badge.
- (especially in the plural as "widow's weeds") (Female) mourning apparel.
- A sudden illness or relapse, often attended with fever, which befalls those who are about to give birth, are giving birth, or have recently given birth or miscarried or aborted.
- Lymphangitis in a horse.
- To urinate.