Ten weeks before Election Day, Gov. Kathy Hochul is undeniably well positioned.

The governor, a Democrat, has a $20 million war chest and the wind at her back in a state where her party’s registered voters outnumber Republicans two to one.

Even so, Ms. Hochul, 67, has something to prove.

Despite New York’s rebound from the pandemic under her leadership, she has struggled to push her approval ratings above 50 percent during her five years in office, as New Yorkers continue to grapple with soaring energy bills, rent and other costs.

A self-described Biden Democrat in a post-Biden landscape, Ms. Hochul has been buffeted with criticism from the left, which is pushing her to do more to tax the wealthy, and from the right, which views her centrist politics as inauthentic and ineffective. The most recent Siena poll showed her just 10 points ahead of her Republican challenger, Bruce Blakeman, the Nassau county executive, in the November election.

In a wide-ranging interview with The New York Times on the eve of her anniversary, the governor reflected on her tenure, which began in the depths of the coronavirus pandemic, and made the case that she deserved some credit for the state’s recovery. “People didn’t know if their state would deliver for them, and I set forth to prove to them that this could actually be a force for good,” she said.

She added, “I believe we have restored confidence in people, in the state of New York.”

When Ms. Hochul was sworn in, the circumstances were sobering. Her predecessor, Andrew M. Cuomo, had resigned in disgrace. Tens of thousands of New Yorkers had died of Covid-19. With violent crime on the rise and remote work becoming the norm, Midtown Manhattan felt like a ghost town.

“People were writing the obituary for this city and this state,” she said.

Today, the picture is very different: Crime is down, employment rates and tourism have largely rallied, and, as the Knicks’ championship victory this summer made clear, people are celebrating again.

As proof of the state’s rebound, Ms. Hochul described a recent Sunday when she visited Coney Island. After a walk down the boardwalk with her husband, the governor stopped into Ruby’s Bar and Grill for a beer.

At the bar, she said, she met a crew of older women in blue dance uniforms, who were incensed that their governor did not recognize them: “‘You don’t know the Pacemakers? We just performed at the Cyclones baseball game next door!’” she recalled with a laugh.

“So I said, ‘All right, let’s turn on the jukebox. Show me what you got,’” Ms. Hochul recalled. Before long, she said, everyone in the bar was dancing.

“That, Ms. Hochul said proudly, “is New York.”

Despite more than a decade in the public eye, including seven years as lieutenant governor, Ms. Hochul has kept a more modest profile than her domineering predecessor, Mr. Cuomo.

Her style is ruthlessly practical and occasionally hokey. And while it has yielded some real results — including her success in resisting some of President Trump’s efforts to push his agenda in New York — she has not been floated as a potential presidential candidate, unlike the governors of some other large blue states.

She often talks about being the first “mom governor,” a title that reflects not only her public persona but also some of her proudest accomplishments: expanding the child tax credit; offering paid prenatal leave; providing free breakfasts and lunches for students; banning cellphones in schools; investing $12 billion to expand access to child care across the state.

Jasmine Gripper, the head of the Working Families Party, said the governor deserved credit not only for the pandemic recovery but also for her commitments on child care and education, and her efforts to protect New Yorkers amid the president’s mass deportation campaign and federal funding cuts.

“The thing that Kathy Hochul has gotten right is really governing for families,” she said. Even on issues where the left-leaning party has butted heads with the governor, Ms. Gripper said, it had found her to be a good-faith negotiator: “The give and take is there.”

But Representative Mike Lawler, a Hudson Valley Republican, said that she had not used her executive power to impose moderation on the state’s left-leaning lawmakers, resulting in a legacy of half measures that had eroded voters’ trust.

“We continue to lead the nation in all the wrong categories: highest tax burden, worst business climate,” Mr. Lawler said, adding: “She responds to the polls. She doesn’t lead.”

The Siena poll showed her with a seemingly narrower lead than in previous polls, though the surveys used different samples. But a narrow majority of respondents said they trusted her on key issues like affordability, public safety, immigration and housing.

“I’m real proud of what we’ve done in public safety,” Ms. Hochul said in the interview, “and I’m going to say this: We’re not done at all.”

Along with the pandemic, Ms. Hochul’s early years in office were dominated by criminal justice issues, on which she found herself at odds with many of her fellow Democrats in the State Legislature.

Ms. Hochul spent years — and considerable political capital — pushing to significantly tighten the bail reform measures that Democrats enacted in 2019 after the death of Kalief Browder, a young man who took his own life after spending three years in jail awaiting trial for stealing a backpack.

The governor argued that the law was too lenient, and inserted provisions to crack down on repeat offenders and give judges and prosecutors more leeway. Along the way she developed her style of dealing with the Legislature: even-keeled, iron-fisted and apparently unbothered by complaints about increasingly late state budgets.

That style has become increasingly evident as she has found herself negotiating with big personalities, from Mr. Trump to Mayor Zohran Mamdani.

When Mr. Mamdani, a 34-year-old democratic socialist, was elected last fall, many expected far less from his relationship with the governor. The two leaders represent different generations and different wings of the Democratic Party and have significant policy and ideological disagreements. But they quickly began working together to chart a course for an expensive but transformative investment in child care.

Mr. Mamdani endorsed Ms. Hochul shortly thereafter. His support helped to doom the campaign of her progressive primary challenger, Lt. Gov. Antonio Delgado. And the mayor and governor reached another major agreement this spring. Facing deep cuts in federal aid and a city budget deficit, Ms. Hochul pushed for a new tax on high-end second homes in New York City in the state budget — allowing Mr. Mamdani to claim a limited victory in his campaign promise to “tax the rich.”

While the governor was less than thrilled with Mr. Mamdani’s rollout of that tax, their partnership has so far proved durable, and some in the party are even looking to it as an example of a possible path forward at a time when the Democratic establishment has been shaken by leftist victories around the country.

Representative Pat Ryan, a moderate Democrat from the Hudson Valley, praised the governor’s collaborative approach. “I think her relationship with Mayor Mamdani is a national model of how to work across a broad Democratic coalition and not agree on every single thing, but focus on where you agree,” he said, adding, “That is very hard to do in today’s politics.”

Indeed, Ms. Hochul has often tangled with her party, particularly the left wing, over issues like taxes and climate policy. Environmental groups and some fellow Democrats were alarmed last year when she moved to weaken the state’s landmark climate law; she has said that the state must adapt to changing economic conditions.

Despite pressure from groups like the Democratic Socialists of America to do more to tax wealthy New Yorkers this year, the governor said in the interview that she had “no intention” of raising taxes or implementing new ones, though she indicated that could change if there were additional unforeseen cuts in federal aid to New York.

But she has recently shown an increased appetite for taking on issues championed by the left, in ways that align with her own priorities.

Last month, she signed an executive order instituting a one-year moratorium on the construction of so-called hyperscale data centers while the environmental, economic and social effects are studied.

“I embrace tech companies,” Ms. Hochul said. But, she added, she expects them to do their part for the state by delivering taxes and jobs.

“If you want to come out, come and show a community you want them to benefit,” Ms. Hochul said. “Help subsidize the utility bills!”