Barely a month after the Food and Drug Administration authorized Covid-19 vaccines for very young children, the prognosis that large numbers of them will actually get the shots looks bleak, according to a new survey of parents released on Tuesday by the Kaiser Family Foundation, which has monitored vaccine attitudes throughout the pandemic.

A majority of parents polled said they considered the vaccine a greater risk to their children than the coronavirus itself.

For children in the age group, 6 months through 4 years, parental apprehension has so far resulted in the administration of scarcely a trickle of Covid shots. Since June 18, when they became eligible, just 2.8 percent of those children had received shots, the foundation found recently in a separate analysis of federal vaccine data. By comparison, 18.5 percent of children ages 5 through 11, who have been eligible for Covid shots since October, had been vaccinated at a similar point in the rollout of their shots.

The new survey found that 43 percent of parents with children under 5 said they would “definitely not” have them vaccinated. About 27 percent said they would “wait and see,” while another 13 percent said they would have their children vaccinated “only if required.” Even some parents who were themselves vaccinated against Covid said they would not give permission for their youngest children.

The new analysis of parents’ views comes as vaccine uptake for older children has been slowing markedly. To date, only 40 percent of children 5 to 11 have been vaccinated. In the new survey, 37 percent of parents said they would “definitely not” get a Covid vaccine for their child in that age group.

The parents’ chief concerns were about potential side effects of the vaccine, its relative newness and what they felt was a lack of sufficient research. Many parents said they were prepared to let their children take the risk of contracting Covid rather than getting a vaccine to prevent it.

Experts on childhood vaccination said they viewed the parents’ hesitation with alarm, coming at a time when Covid cases are once again soaring and expected to worsen during the cold weather months, and as the possibility of new and potentially more dangerous coronavirus variants remains.

Although a vast majority of children who come down with Covid get over it easily, “some kids get very, very ill from it and some die,” said Patricia A. Stinchfield, the president of the National Foundation for Infectious Diseases. She was not involved in the Kaiser study.

How a child will fare with Covid is unpredictable, added Ms. Stinchfield, a nurse practitioner who coordinated vaccine administration for Children’s Minnesota, a children’s hospital system in St. Paul and Minneapolis. “We have no marker for that,” she said. “Half the kids who come down with severe Covid are healthy kids, with no underlying conditions. So the idea of saying ‘I’m going to skip this vaccine for my kid, we’re not worried about Covid’ is really to take a risk.”

This latest report is based on an online and telephone survey from June 7 to June 17 of 1,847 adults, 471 of whom had a child under 5. The margin of error was plus or minus 3 percentage points for the full sample, and plus or minus 8 percentage points for parents with a child under 5.

Perhaps unsurprisingly, the partisan divide was especially sharp around vaccination for children, with Republican parents three times as likely as Democratic parents to say they will “definitely not” have their child vaccinated.

A majority of parents said they found information from the federal government about the vaccine for their children to be confusing. Yet 70 percent said they had not yet discussed the shots with a pediatrician. Just 27 percent of those parents who are considering the vaccine said they would make an appointment to have that conversation.

Parents who might be predisposed to having their children get Covid shots said that lack of access was a significant barrier, a concern expressed by more Black and Hispanic parents than white parents. About 44 percent of Black parents worried about having to take time off from work to have their children vaccinated or to care for them if the children had side effects. Among Hispanic parents of young children, 45 percent said they were worried about finding a trustworthy location for the shots, and about a third feared they would have to pay a fee.

Ms. Stinchfield said she understood their concerns: Her own daughter had to take off work to get vaccinations for Ms. Stinchfield’s grandchildren, ages 1 and 3. Ms. Stinchfield went to a clinic with them. “The message to clinics is, Make the vaccine for kids available in the evenings and on weekends,” she said.

Did her grandchildren have any side effects? No, Ms. Stinchfield said with a chuckle. “They felt so good that we put them in a little kiddie pool,” she said. “And now my granddaughter’s got a tan line from the Band-Aid from the shot on her leg.”