The most common and most likely answer to this crossword clue is the 4 letter word JESS
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More clues leading to the result JESS
Buddy explains!

The straight part
We know that "Restraining strap" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case restraining strap = jess.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 JESS mean?
JESS noun- A short strap fastened around the leg of a bird used in falconry, to which a leash may be fastened.
- To fasten a strap around the leg of a hawk.
- A collimated stream, spurt or flow of liquid or gas from a pressurized container, an engine, etc.
- A spout or nozzle for creating a jet of fluid.
- A type of airplane using jet engines rather than propellers.
- An engine that propels a vehicle using a stream of fluid as propulsion.
- A part of a carburetor that controls the amount of fuel mixed with the air.
- A narrow cone of hadrons and other particles produced by the hadronization of a quark or gluon.
- Drift; scope; range, as of an argument.
- The sprue of a type, which is broken from it when the type is cold.
- A hard, black form of coal, sometimes used in jewellery.
- (color) The colour of jet coal, deep grey.
- Only, simply, merely.
- (sentence adverb) Used to reduce the force of an imperative; simply.
- Used to convey a less serious or formal tone
- Used to show humility.
- (degree) absolutely, positively
- Moments ago, recently.
- By a narrow margin; closely; nearly.
- Exactly, precisely, perfectly.