Linguistic Term for "Shame if something happened to it" type phrases

The phrase "you've got a nice [noun] here. Shame if something happened to it" has been a trope for years. It represents a type of communication where what the speaker intends to communicate is not actually what they say.

Is there a linguistic term for this? Subtext doesn't seem to fit because that implies that it is a secondary meaning, rather than the primary intended meaning.

The common way to describe this is as veiled language.

veiled 1 adj A veiled comment is expressed in a disguised form rather than directly and openly.

• This last clause is a thinly-veiled threat to those who might choose to ignore the decree.

[Collins]

implicature

n. The action of implying a meaning beyone the literal sense of what is explicitly stated, e.g., saying the frame is nice and implying I don't like the picture in it.
■ a meaning so implied. (The New Oxford American Dictionary)

An implicature is something the speaker suggests or implies with an utterance, even though it is not literally expressed. Implicatures can aid in communicating more efficiently than by explicitly saying everything we want to communicate. This phenomenon is part of pragmatics, a subdiscipline of linguistics. The philosopher H. P. Grice coined the term in 1975. Grice distinguished conversational implicatures, which arise because speakers are expected to respect general rules of conversation, and conventional ones, which are tied to certain words such as "but" or "therefore". Wiki

Consider the following example (lifted unashamedly from Steven Pinker’s book The Stuff of Thought):

A gangster walks into a local restaurant. The restaurant has been doing well recently, and the local criminal gangs are aware of this fact. The gangster walks over to the restaurant owner, stares conspicuously around the room, and says “This is real nice place you got here. It would be a shame if something happened to it.”

Ostensibly, the gangster’s statement is one of fact: depending on what the “something” in question is, it may indeed be a shame if it happened to the restaurant. But of course no one reading the statement really thinks it is as innocuous as that. Everyone knows that it constitutes a thinly-veiled threat. Why is this?

The answer lies in something known as conversational implicature, which is the fancy label given to the mundane phenomenon that the semantic content of a particular utterance or sentence is not exhausted by the meaning of the words that make up that utterance. Which is to say: it is possible for an utterance to have an implied meaning, which is just as important and just as readily understood as that of its explicit meaning. Indeed, sometimes it is more important than the explicit meaning, as in the case of the gangster’s veiled threat: If the restaurant owner didn’t pick up on the implied meaning, he could create problems for himself. "Implicature and the Interpretation of the Law (Part One)"

Issuing a warning or else ​"that something unpleasant might happen to them" is a Thinly Veiled Threat.

Examples

• Not for its political points: The New York Times reported seriously that President Trump's threats to withdraw from NATO sent officials scrambling to prevent the alliance from turning into a disaster. They did not see the thinly veiled threat as posturing ... for countries to pay their proportional NATO dues as the US did. [rephrased]

  But seen as a veiled threat, Trump's suggestion to withdraw was a warning, only: I don't want to play in your yard, if you won't be good to me. And if you can't bluff, you can't bargain.

• Even vanilla comments like "You got a nice family there" can take on sinister tones when serving as the punchline to a demand with hints of violence.