An influx of new revenue has reopened the District’s budget debate, with Mayor Vincent C. Gray pledging to increase police staffing and maintain funding for school nurses. But some D.C. Council members are continuing to question Gray’s late requests to increase funding for city Medicaid contractors.

Chief Financial Officer Natwar M. Gandhi announced last week that the District is expected to take in $107 million more than expected in the current fiscal year, which ends Sept. 30. In a briefing for council members Wednesday, Gray said he planned to set aside $10.8 million to hire enough police officers to keep the city force at 3,900.

Without the new funds, the force would drop to below 3,800 by the end of the next fiscal year — a level Police Chief Cathy L. Lanier has said will lead to “trouble.”

Much of the remainder of the influx will offset overspending and unanticipated costs in city agencies. Gray said he will submit a modified budget to the council later this week. There, members could make further changes. Jack Evans (D-Ward 2) said he would take the opportunity to renew his efforts to eliminate a new tax on non-District municipal bonds, which had previously been exempt from city income tax.

“I’m going to try,” he said. “If we’re taking care of all this other stuff, we ought to be able to take care of this.”

At Wednesday’s briefing, questions persisted about Gray’s request earlier this month to send an additional $32 million to the two companies that manage the city’s Medicaid program.

The concerns were amplified by news that the city has agreed to pay one of them $10.2 million to settle a contract dispute, contributing to the funding boost.

In September, D.C. Chartered Health Plan petitioned the Contract Appeals Board for $14.9 million in compensation for the city’s “failing to set actuarially sound rates” for dental care. The dispute is rooted in a court settlement in which the District pledged to improve children’s access to dental care. The city, according to Chartered’s appeal, ordered the provider to dramatically increase its reimbursement rates for dentists so more would operate in the city and serve needy kids. That “exacerbated existing pressures on Chartered’s ability to control costs,” the appeal said.

In October, the city moved to dismiss on procedural grounds without addressing the merits of Chartered’s argument. The board did not rule on the motion before the settlement was reached.

Health Care Finance Director Wayne Turnage said Tuesday that the District was compelled to settle the complaint ahead of the city’s annual negotiation with the two Medicaid managed-care providers, of which Chartered is the largest. He said that actuaries employed by the city estimated the District’s “exposure” at $10 million. But David A. Catania (I-At Large) argued Wednesday that managed-care providers are expected to assume the risk that the rates they agree to might not cover their costs. The settlement, he said, amounted to a “retroactive rate increase.”

As part of the settlement, the District has agreed to pay Chartered a lump sum of $3.7 million. It also agreed to pay a premium over the monthly per-enrollee rates it is offering its other contractor, UnitedHealth, for an additional $6.5 million.

A Chartered spokeswoman did not return a call for comment. The company has contracted with the city for more than two decades, and its chairman, Jeffrey Thompson, enjoys close political ties to city officials. The city sued the company over improper billings in 2008, resulting in a $12 million settlement.

“We make them entirely whole in one fell swoop,” Catania said about the new settlement. He called the Gray administration’s Medicaid numbers “silly.”

He was joined in questioning the contracts by Jim Graham (D-Ward 1). “We need to be suspicious of this $32 million,” he said. “It should be less than that.”

Staff writer Nikita Stewart contributed to this report.