The most common and most likely answer to this crossword clue is the 4 letter word SUPS
I've seen this crossword clue in The Daily Telegraph
S
U
P
S
More clues leading to the result SUPS
Buddy explains!

The straight part
We know that "Starts to swallow using, perhaps, spoon" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case starts to swallow using, perhaps, spoon = sups.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 SUPS mean?
SUPS noun- A sip; a small amount of food or drink.
- Superintendent.
- Upper limit.
- A stand-up paddleboard