The most common and most likely answer to this crossword clue is the 4 letter word SHED
I've seen this crossword clue in The Times
S
H
E
D
Buddy explains!

The straight part
We know that "Simple building/left behind" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case simple building/left behind = shed.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 SHED mean?
SHED verb- To part, separate or divide.
- To part with, separate from, leave off; cast off, let fall, be divested of.
- To pour; to make flow.
- To allow to flow or fall.
- To radiate, cast, give off (light); see also shed light on.
- To pour forth, give off, impart.
- To fall in drops; to pour.
- To sprinkle; to intersperse; to cover.
- To divide, as the warp threads, so as to form a shed, or passageway, for the shuttle.
- An area between upper and lower warp yarns through which the weft is woven.
- A distinction or dividing-line.
- A parting in the hair.
- The top of the head.
- An area of land as distinguished from those around it.
- A slight or temporary structure built to shade or shelter something; a structure usually open in front; an outbuilding; a hut.
- A large temporary open structure for reception of goods.
- An automobile which is old, worn-out, slow, or otherwise of poor quality.
- (rail transportation) A British Rail Class 66 locomotive.
- To place or allocate a vehicle, such as a locomotive, in or to a depot or shed.