The most common and most likely answer to this crossword clue is the 4 letter word FELL
I've seen this crossword clue in The Times
F
E
L
L
Buddy explains!

The straight part
We know that "Went down/hill" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case went down/hill = fell.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 FELL mean?
FELL noun- A cutting-down of timber.
- The stitching down of a fold of cloth; specifically, the portion of a kilt, from the waist to the seat, where the pleats are stitched down.
- The end of a web, formed by the last thread of the weft.
- To make something fall; especially to chop down a tree.
- To strike down, kill, destroy.
- To stitch down a protruding flap of fabric, as a seam allowance, or pleat.
- An animal skin, hide, pelt.
- Human skin (now only as a metaphorical use of previous sense).
- (obsolete outside Britain) A rocky ridge or chain of mountains.
- (obsolete outside Britain) A wild field or upland moor.
- Of a strong and cruel nature; eager and unsparing; grim; fierce; ruthless; savage.
- Strong and fiery; biting; keen; sharp; pungent
- Very large; huge.
- Eager; earnest; intent.
- Sharply; fiercely.
- Anger; gall; melancholy.
- The finer portions of ore, which go through the meshes when the ore is sorted by sifting.
- (heading) To be moved downwards.
- To move downwards.
- To happen, to change negatively.
- To be allotted to; to arrive through chance, fate, or inheritance.
- To diminish; to lessen or lower.
- To bring forth.
- To issue forth into life; to be brought forth; said of the young of certain animals.
- To descend in character or reputation; to become degraded; to sink into vice, error, or sin.
- To become ensnared or entrapped; to be worse off than before.
- To assume a look of shame or disappointment; to become or appear dejected; said of the face.
- To happen; to come to pass; to chance or light (upon).
- To begin with haste, ardour, or vehemence; to rush or hurry.
- To be dropped or uttered carelessly.
- (of a fabric) To hang down (under the influence of gravity).