And although her tenure in the military and in the de Blasio administration is generally well regarded, she did suddenly resign as the director of the Defense Centers for Excellence in 2010, after news reports and congressional criticism that the military was failing to diagnose and treat soldiers with mild traumatic brain injuries.

That has not stopped some people from signing on to her campaign, which hopes to spend $7 million, the maximum allowed under the city’s matching funds program, for the 2021 primary. Tucker Green, founder of Tucker Green Consulting, and a well-known fund-raiser, has signed on as a consultant.

Ken Fisher, a managing partner of Fisher Brothers, a family-owned real estate development and management firm with $8.5 billion in market valuation, is one of Ms. Sutton’s early financial backers.

Mr. Fisher said he met Ms. Sutton through his work as chief executive of the Fisher House Foundation, which houses veterans being treated for illnesses away from home, and as co-chairman of the Intrepid Sea, Air & Space Museum.

“Being a general is like being a C.E.O.: She’s good at working under pressure,” Mr. Fisher said. “She can manage the city and get along with the governor. Instead of demonizing businesses and corporations as a whole, she sees the value of bringing the public and private together.”

In her 30-year career, Ms. Sutton was the Army’s highest-ranking psychiatrist, completed a tour in Egypt and then went to Iraq during the first gulf war. She helped expand the city’s veterans’ affairs office into a full-blown mayoral agency — the city’s first new agency in two decades — and advocates say she helped drive down chronic homelessness among veterans.

Unlike Mr. de Blasio, who describes himself as a progressive Democrat, Ms. Sutton, a lifelong independent who recently registered as a Democrat, is a moderate.