Over the past six years, thousands of people in Boston who claimed the residential exemption on their property taxes did not qualify for those savings — in some cases, thousands of dollars annually — city records show.

Even worse: There’s nothing the city can do to recover those funds.

Many of those homeowners had up to a $2,000 exemption on their annual property tax bills.

The city’s assessing commissioner, Ronald Rakow, said that under current law, “there is no way to recoup the money’’ from an incorrectly applied exemption.

The city’s only recourse, per state law, is simply to remove the exemption for future taxes. Since 2010, the city has removed the residential tax exemption from 6,604 homeowners, according to assessing information the Globe obtained in a public records request.

“That is [the] concern that the administration has right now, that while we can take it away for future years, we can’t go back,’’ Rakow said.

The residential tax exemption, a flat dollar amount, is given to qualified Boston residents who own and occupy their property. The exemption got a boost — and attracted controversy — in December, after Mayor Martin J. Walsh announced Boston would increase the city’s residential tax exemption to 35 percent of a home’s assessed value. It had been 30 percent.

Rakow said that over the past several years, the city has been urging state lawmakers to give Boston the power to get back the exempted amounts. Walsh is again pressing the Legislature to act, but it is unclear whether his efforts will get any traction.

Rakow said assessing officials conduct an extensive review at the start of each fiscal year to ensure “that the people who receive the exemption deserve the exemption.’’

The analysis includes verifying a person’s primary address.

New homeowners are required to reapply for the exemption held by the seller. Officials flag parcels that raise concerns for a more extensive audit.

Since 2010, city officials have reviewed the residential exemptions for 76,211 parcels. Officials then selected 9,204 parcels for a deeper audit. They asked the owners of 7,904 parcels to resubmit an application for the exemption. And just 1,300 of them were verified, the data show.

Last year, the Assessing Department audited 1,099 properties and found 938 where residency could not be verified. Just 161 of those eventually qualified, the data show.

Rakow said very few of the cases identified by his staff involved a property owner trying to game the system. But the Assessing Department does not record the reason a property owner’s tax exemption is selected for additional review.

“I would say that it’s infrequent that we get an application signed by somebody under the pains and penalties of perjury saying that they live at the address and it turns out that they don’t,’’ Rakow said.

“There may be other things involved, like maybe the property is in a trust or there may be other conditions where they don’t qualify,’’ he said.

Rakow said the auditing — performed in the months before the third-quarter tax bill is sent in December — is conducted before the exemption is granted, keeping such cases to a minimum.

Walsh’s recent property tax changes, which followed a change in state law, meant that qualified owners of single-family properties would see a bigger share of their tax bills, $2,433,exempted from taxes this year.

Last year, it was about $1,962. The average property tax bill in Boston is over $3,000.

The exemption equates to a property value reduction of about $229,000, said Samuel R. Tyler, the president of the Boston Municipal Research Bureau.

“If your home is worth $400,000 and you subtract $229,000, then you are being taxed on what’s left — and that’s $171,000,’’ Tyler said.

“This a tax break that is intended to help retain the middle class in Boston,’’ he said.

City officials have hailed the effort as needed relief for residents strained by stagnated wages and higher living expenses.

But Walsh administration officials came under fire after it became clear they were shifting the tax burden onto owners of large multi-unit dwellings who do not live in their properties. Owners of high-valued homes, apartment buildings, and properties not occupied by the homeowner will pay more, officials acknowledged.

Critics said that by increasing the residential exemption, the city would be losing out on needed revenue that could have gone to other services — an assertion that Tyler and Boston officials refuted.

To offset any losses, Boston officials increased the residential tax rate, which allows them to raise the full amount of residential property tax revenue that’s allowed under the law.

Critics also raised the possibility of people claiming the tax exemption for a property when they live elsewhere.

City Councilor Annissa Essaibi-George, whom the Globe asked to review the city data, said she has concerns about erroneous exemptions.

“If a person is misrepresenting their home address to qualify for an exemption, we need to not only remove the exemption, but also complete an extensive audit to better understand the financial benefit that property owners received because of the misrepresentation,’’ she said.

Essaibi-George said the city should seek repayment of the amount exempted, plus potential fines.

Meghan E. Irons can be reached at meghan.irons@globe.com. Follow her on Twitter @meghanirons.