The city Department of Health & Mental Hygiene has sunk its claws into another beloved New York institution — The Algonquin hotel’s lobby cat.

Matilda III — the latest in an illustrious line of free-roaming Algonquin felines — has been banished from the lobby lounge, leaving guests fruitlessly searching for her under chairs and sofas.

Prodded by Nanny Bloomberg, the DOH has been socking restaurants with steep fines for minor violations — and slapping dreaded “C” ratings on places where no one was known to get sick.

Some places are taking no chances, eliminating popular features before the DOH can strike them down. The party-pooping agency recently nudged Sardi’s to eliminate cheese snacks at its bar.

Now, thanks to a DOH “reminder,” poor Matilda is on a leash behind The Algonquin’s check-in desk, or out of sight on a higher floor.

The city’s favorite feline, a blue-eyed ragdoll, took up residence last winter. She’s the 10th Algonquin cat since Rusty, a k a Hamlet I, moved into the hotel, legendary home of the “Round Table” literary salon, in 1932.

The pampered pussies are as much a part of The Algonquin’s cozy confines as the oak paneling and upholstered chairs and sofas. Hotel staff have delighted in pointing out Matilda’s hiding places to guests.

Like her predecessors, she had the run of the house, but the lobby, home to the Round Table restaurant and lounge, was called her “natural habitat.”

Hotel staff tried blaming Matilda’s going missing on guests who were abusing her — a claim that gave paws, since it was not a problem for 80 years.

Algonquin General Manager Gary Budge first said the lockup was for Matilda’s own good.

“People seem more aggressive toward her, and she’s responding in a way that’s not helpful,” he claimed. But then he acknowledged, “The [Health] department in the past months suggested to us that pets in food-service facilities are no longer commingled.

“The lobby is an area where we serve food and beverage. We always want to be respectful of the Department of Health.”

A DOH spokeswoman said, “According to the New York City Health Code, live animals are not allowed in food-service establishments unless a patron needs a service dog.”

She said Algonquin managers told an inspector they had an “electronic fence to contain the cat,” and the inspector “reminded them that the cat is not allowed in food-service or preparation areas.”

At a preliminary inspection on Nov. 10, The Algonquin received 20 violation points, worth a B grade if they aren’t corrected on a follow-up DOH visit.

None of the violations involved Matilda.

The hotel will close Jan. 1 for a four-month, $15 million renovation.