President Biden threw New York a dead mackerel this week, and maybe — just maybe — the folks running the Empire State are beginning to get Washington’s message.

Which seems to be: We don’t care about border integrity, and you folks are just going to have to live with the consequences.

Specifically, the White House Wednesday announced that some Venezuelans among the tens of thousands of border-crossers packed into the state will be eligible for an expedited work-permit process.

Mayor Adams’ reaction was typical: He basically has bet the ranch on work permits, and he seemed to think that migration salvation is at hand:

“Those entering our city and our country can provide for themselves and finally have a shot at the American Dream.”

And Gov. Hochul chimed right in — terming the development a “good first step.”

Then reality – or was it some scary polling? – came to the fore: Hochul, bless her Erie County heart, said maybe the Big Apple’s “right-to-shelter” policy is a migrant magnet, and that maybe something should be done about it.

Time will tell what she has in mind — but she’s finally approaching the right track.

So much so, it appears, that Adams himself chirped a policy pronouncement: He wants to limit shelter stays to 30 days for single adults, and 60 days for families.

As it stands, anyone who can stumble into the state gets three meals daily plus a cot, as of right, and there’s a strong constituency for this.

So whether Hochul and Adams have the skills to buck the radicals running the city council; New York’s relentlessly liberal appeals courts; and the fat-cat not-for-profit operators making bank off the city’s robust shelter industry is an open question.

Ditto, that they’re even inclined to try.

It should be beyond dispute, however, that Adams’ “American-Dream-for-the-world” notion is fiscal, social and cultural nonsense.

Indeed, news also arrived this week of some 11,000 border-crossers jamming into Eagle Pass, Texas (population 29,000); it’s safe to bet that most of them have their sights squarely set on Manhattan’s Roosevelt Hotel.

And, never mind her second thoughts, Hochul’s initial reaction was only marginally more sensible than the mayor’s.

She took to television Wednesday to declare that New York is “at capacity” – adding “We have to let people know that if you’re thinking of coming to New York, we are truly out of space.”

Ah, but to someone bobbing in the water, it naturally seems like there’s room in the lifeboat for at least one more — especially when Hochul is waving border-crossers aboard by promising “to ensure they are connected to jobs as soon as they’re able to work.”

Can we say “mixed message,” governor? Mr. Mayor?

To be fair, of course, Hizzoner has had a rough week.

The president was in town for the UN’s annual Midtown traffic jam, but couldn’t find five minutes for him. And then the administration flipped him the bird when he invited Biden to visit the Roosevelt Hotel.

So, clearly, Adams gets no respect.

It’s up to him to earn it from New Yorkers, starting with his extremely tentative retreat on right-to-shelter.

But little steps for little feet are better than no steps at all.

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