The Federal Housing Administration said Friday it will once again raise its mortgage-insurance premiums and take a series of other steps in hopes of putting the agency on firmer financial footing and avoiding the need to seek taxpayer aid for the first time in its 78-year history.

The move would mean new borrowers of FHA-backed loans would pay an average of $13 more per month in premiums. While the increase should help to bolster the agency’s bottom line, every bump in prices also makes mortgages more expensive for the very borrowers that the FHA is tasked with helping.

“It’s a tightrope,” FHA commissioner Carol Galante said Friday of carrying out her agency’s mission while also ensuring it stays solvent. “We continue to look at that balance every day.”

The changes come as an independent audit of the FHA released Friday showed that mounting losses from mortgage delinquencies — particularly on loans the FHA backed between 2007 and 2009 — left the agency’s reserves with a projected $16.3 billion deficit as of Sept. 30.

Friday’s results underscored the debate about the proper role of the government in backstopping the nation’s housing market and how much risk taxpayers should face in order to ensure that mortgage markets continue functioning through good times and bad. Roughly nine out of 10 new mortgages are backed by the federal government.

The FHA, created in the wake of the Great Depression to help revive the nation’s housing market by providing the sort of insurance that would encourage banks to lend again, currently insures more than $1 trillion in mortgages. Its share of the mortgage market spiked during the height of the housing crisis but has declined somewhat in recent years.

Housing and Urban Development Secretary Shaun Donovan expressed confidence that the FHA’s finances will improve over time, saying Friday’s troubling projections grew out of the outsized role that FHA found itself playing during the economic downturn.

“With its dual mission of providing access to homeownership for underserved populations and supporting the housing market during tough times, there’s little doubt that FHA helped prevent a much deeper crisis,” Donovan said. “That progress, however, has not been without stress.”

Critics such as Sen. Bob Corker (R-Tenn.) say the FHA has “strayed a long way from its original mission” and that the government should not be in the business of backing loans “to people who can’t pay them back.”

Edward Pinto, a resident fellow at the conservative American Enterprise Institute who has long warned about FHA’s deteriorating financial condition, wrote that Friday’s report “should be cause for significant concern for Congress and taxpayers.” He and other analysts say the agency’s fiscal health may get worse before it gets better and added that it should be much more judicious about the loans it chooses to insure going forward.

FHA officials were quick to point out that Friday’s projections don’t take into account recent improvements in the housing market or revenues the agency is expected to generate in the future, nor do they suggest that the FHA has an immediate cash flow problem. The audit was conducted by Integrated Financial Engineer Group in Rockville.

In addition, the agency said this year’s annual audit used more conservative projections for home price appreciation than a year earlier, leading to less optimistic estimates. Officials said that historically low interest rates, while positive for the overall economy, have cost the FHA revenue because borrowers often refinance or pay off their loans, depriving the agency of premiums.

Even so, if the current projected shortages were to persist, the FHA could need an infusion of cash from the Treasury Department as soon as next fall. That determination would be made by the Obama administration and would not require congressional approval.

Officials said Friday they hope to stave off that possibility. In addition to the increase in premiums, which amounts to one-tenth of a percent, the agency is seeking to streamline short sales on homes financed by FHA loans, tightening some lending standards and seeking expanded authority from Congress that would give the agency more flexibility to tweak loan programs and minimize losses.