The election interference case against former President Donald J. Trump and 14 of his allies in Georgia has been buffeted by allegations that the main prosecutor, Fani T. Willis, engaged in a romantic relationship with a lawyer she hired to help with the case.

Mr. Trump and one of his co-defendants, Michael Roman, are now trying to convince a judge that the alleged relationship has created a conflict of interest that should be grounds for disqualifying Ms. Willis, the Fulton County district attorney; the hired lawyer, Nathan J. Wade; and Ms. Willis’s entire office.

Though the legal question before the judge will not be taken up until mid-February, some important developments in the matter may unfold this week. Mr. Wade has been ordered to appear at a hearing regarding his divorce case on Wednesday, when he could be forced to take the stand and field questions about his relationship with Ms. Willis. And Ms. Willis is facing a Friday deadline for her legal response to the effort to have her thrown off the election interference case.

The accusations were first made in a Jan. 8 court motion filed by Mr. Roman, a former Trump campaign official. Both Mr. Trump and Mr. Roman are also seeking to have the cases against them dismissed, although these efforts are likely long shots.

The question of whether the prosecutors should be disqualified if the claims are proven true has divided legal experts. Some have called on Mr. Wade and Ms. Willis to resign from the case, in part to avoid the possibility of delaying a trial date.

No matter how the judge rules, the allegations have become a distraction for Ms. Willis, a Democrat and veteran prosecutor who is up for re-election this year. Investigations into her conduct are likely to gear up soon, led by panels controlled by Republicans who enjoy healthy majorities in the Georgia legislature.

Here are a few things to know about this complicated subplot in the prosecution of Mr. Trump and others for trying to overturn his 2020 election loss in Georgia.

What are the accusations against Fani Willis?

In the court filings seeking to throw her off the case, Ms. Willis is accused of having hired Mr. Wade, who had never led a high-profile criminal prosecution, because she was in a romantic relationship with him.

She is also accused of having profited from Mr. Wade’s earnings from her office — taxpayer funds — by going with him on vacations that he sometimes paid for.

Mr. Roman’s motion notes that Georgia law recognizes conflicts of interest, particularly those in which a prosecutor is shown to have “a personal interest or stake in the defendant’s conviction,” as grounds for removal.

In this case, the motion says, Mr. Wade was paid more than $650,000 since starting his job with the district attorney’s office in November 2021. The filing claims that he subsequently paid some of the costs of vacations that he and Ms. Willis took together to Napa Valley, Calif., Miami and the Caribbean.

Excerpts from Mr. Wade’s credit card statements, released by his estranged wife as part of their divorce case, show that he has bought airline tickets for himself and Ms. Willis to Miami and San Francisco.

All of this, Mr. Roman says in the filing, amounts to Ms. Willis awarding “lucrative contracts to her boyfriend” and then benefiting from them. As a result, the filing states, “both of them have acquired a personal interest and stake in Mr. Roman’s conviction, thus depriving Mr. Roman of his right to a fundamentally fair trial.”

The motion claims that the two lawyers began their relationship before Mr. Wade was hired, and it argues that Mr. Wade was underqualified to handle such a big case. Reporting from The New York Times did not turn up any evidence of Mr. Wade having ever prosecuted a major political corruption case or criminal racketeering case.

Mr. Roman also accuses Ms. Willis of violating county laws and state bar rules, and possibly federal laws concerning honest services fraud and racketeering.

A separate filing from Mr. Trump on Jan. 25 argued that Ms. Willis violated Georgia bar rules for prosecutors when she claimed during a recent speech at an Atlanta church that racism was one of the reasons she was under scrutiny. Both Ms. Willis and Mr. Wade are Black.

Mr. Roman’s motion also seeks to dismiss the indictment against him, on the grounds that Ms. Willis lacked the legal authority to appoint Mr. Wade as a special prosecutor, because her office did not seek the approval of Fulton County’s board of commissioners in hiring Mr. Wade. But both the chair of the commission and the county attorney have said that Ms. Willis did not require county approval to hire Mr. Wade.

What has Ms. Willis said about the allegations?

Ms. Willis has not addressed the allegations of her having a romantic relationship with Mr. Wade. Her office has said that the response will come in formal legal filings. In her speech at the Atlanta church, Ms. Willis referred to herself as an “imperfect” person.

Addressing God, she said, “You did not tell me, as a woman of color, it would not matter what I did — my motive, my talent, my ability and my character would be constantly attacked.”

What kind of problems could this pose for the election interference case?

It seems unlikely that the allegations will cause the Trump case to be dismissed. But they may already be influencing the way future jurors think about Ms. Willis’s integrity and the prosecution team.

The presiding judge in the case, Scott McAfee of Fulton County Superior Court, will have to rule on Mr. Roman’s motion to disqualify the prosecutors, and has set a hearing for Feb. 15.

While disqualification might seem like an extreme measure, there is precedent within the Trump case itself. In July 2022, Judge McAfee’s predecessor, Robert C.I. McBurney, disqualified Ms. Willis and her office from developing a criminal case against Burt Jones, now Georgia’s lieutenant governor, because Ms. Willis had headlined a fund-raiser for one of his political opponents.

If Ms. Willis’s office were to be taken off the Trump case, the Prosecuting Attorneys’ Council of Georgia, a state agency, would have to find another prosecutor. That group has yet to find a prosecutor to take up the Jones case.

One of the most prominent legal figures to reject the idea that Ms. Willis should be disqualified is Norman Eisen, a senior fellow at the Brookings Institution who served as special counsel to the House Judiciary Committee during the first Trump impeachment.

Mr. Eisen, along with the legal scholars Richard Painter and Joyce Vance, wrote in a recent article that the allegations against the two prosecutors “are as irrelevant to the trial as allegations in other situations that prosecutors took office supplies for personal use, drove county vehicles for personal errands, or plagiarized portions of their student law review notes.”

“All of those are legitimate issues — for prosecutors’ offices and those with oversight responsibilities to address — but such allegations do not bring criminal prosecutions to a stop or require that cases be transferred to a different office,” they wrote.

What could the repercussions be for Ms. Willis and Mr. Wade?

The two prosecutors will probably face numerous investigations. Representative Marjorie Taylor Greene, Republican of Georgia and a Trump ally, has filed a complaint with the state ethics commission. The State Bar of Georgia has probably also received complaints, although such complaints are not public record at the time they are filed.

Mr. Wade and Ms. Willis are also likely to be investigated by a state government body created last year called the Prosecuting Attorneys’ Qualifications Commission. Though currently in a legal and operational limbo, the commission may be up and running in the coming weeks or months. It is controlled by Republicans and will most likely have the power to punish or remove Georgia prosecutors.

Separately, the Republican-controlled Georgia Senate has created a panel with the express purpose of investigating Ms. Willis. This body has no power to remove her from office, but it has subpoena powers. Its hearings could serve to embarrass and politically harm the district attorney.

State Representative Charlice Byrd, a Republican, has filed a resolution calling for Ms. Willis’s impeachment, but Republicans do not have enough votes in the Senate to unilaterally remove her from office.

Bob Ellis, a Republican on the Fulton County board of commissioners and chair of its audit committee, has begun an inquiry and asked Ms. Willis to provide numerous documents, including invoices and payments to special prosecutors. The commission does not have the power to remove the elected district attorney.

What’s the latest in the case against Donald Trump and his allies?

An indictment handed up by a grand jury in August charged Mr. Trump and 18 of his allies under the state racketeering law, along with other serious crimes. Four defendants have since pleaded guilty. A trial date has not yet been set.