Veterans favor Donald Trump over Democrat Hillary Clinton by 9 percentage points in a new poll released Thursday, though that gap is less than half the margins rolled up by Republicans among that group in the last two elections.

Former military members favored Mr. Trump over Mrs. Clinton by 47 percent to 38 percent, with 14 percent undecided. A whopping 59 percent said they have an unfavorable opinion of the former secretary of state, compared with 47 percent who view Mr. Trump negatively.

The survey comes as the woes at the Veterans Affairs Department flared up again this week, with some ill-advised words from Secretary Robert McDonald and an acknowledgment Wednesday that the agency has wrongly declared more than 4,200 veterans dead in the past four years, disrupting benefits to veterans and their dependents.

The errors were revealed in a letter from the VA to Rep. David Jolly, Florida Republican, who requested the information following a string of mistaken death cases in the Tampa region.

“These numbers confirm our suspicion, that mistaken deaths by the VA have been a widespread problem impacting thousands of veterans across the country,” Mr. Jolly said. “It’s a problem that should have been addressed years ago, as it has caused needless hardships for thousands of people who had their benefits terminated and their world turned upside down.”

Mr. Jolly is a candidate for a Senate seat in Florida this year. He said his congressional staff has helped several veterans get benefits restored after they were wrongly declared dead in the past two years.

The survey of more than 1,600 former military members by Morning Consult, a nonpartisan media and technology company, could be good news for Mr. Trump, who regularly says on the stump that veterans are treated “horribly” by the VA and the Obama administration. But Mr. Trump also has drawn fire from some veterans group for slow or nonexistent follow-through on promises to give “millions” to veterans charities.

Votes of veterans will be a major factor in such swing states with large populations of current and former military members like Virginia, North Carolina, Colorado and Florida.

But veterans’ support in those states is no guarantee of victory; Republican Mitt Romney won only North Carolina among those four states in 2012, despite beating President Obama by about 20 percentage points among veterans nationwide.

Veterans voted for Republicans by a 20-percentage-point margin over Democrats in the midterm House races in 2014, and made up 17 percent of all voters nationally two years ago.

In another sign of soft support among former military members, veterans told Morning Consult pollsters they would prefer the Republican over Sen. Bernard Sanders of Vermont, an avowed socialist, by an even smaller margin than the nine-point gap the tycoon holds over Mrs. Clinton.

In a hypothetical head-to-head matchup with Mrs. Clinton’s remaining rival, veterans favored Mr. Trump over Mr. Sanders by just 47 percent to 43 percent, with 10 percent undecided.

The polling concluded Monday, the same day that Mr. McDonald created a firestorm by comparing veterans’ wait times for health care services to waiting in line at Disneyland.

In the survey, 51 percent of veterans said medical care at VA facilities is worse than the services they could receive from a private doctor or hospital, and only 16 percent said the VA facilities are better. About one in five veterans said the care is the same no matter where they go.

Sixty-three percent of the veterans in the poll said the VA’s problems are “due to long-standing government bureaucracy,” while 21 percent blamed the Obama administration for the agency’s problems.

On the issue of dead veterans, the VA acknowledged that it terminated benefits for 4,201 veterans who weren’t dead from 2011 to 2015, and subsequently reinstated the monthly benefits.

Danny Pummill, a top VA official, said the agency’s computers do not collect information on the cause of the errors, but noted in the agency’s defense that the “accuracy [rate] of award terminations due to death was 99.8 percent.”

The VA announced in December a new process to avoid wrongly terminating benefits of veterans who are mistakenly declared deceased. Under the new guidelines, beneficiaries who are incorrectly declared dead can correct the error with the department before monthly benefits are terminated.

Mr. Jolly said he will ask the department for a new annual survey on the mistakes.

“If the VA’s new policy is indeed working, this problem should be eliminated,” he said. “If the problem persists, then Congress will demand further action. This creates tremendous financial hardships and undue personal turmoil for veterans, many who are seniors relying primarily if not solely on their VA benefits.”

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