It’s been another rough week for Eric Adams, who hopes better ones are coming.  An apparently unprecedented four federal investigations should come to a head by year’s end.  The two U.S. attorneys pursuing those public probes will likely fish or cut bait ahead of Justice Department guidelines against bringing charges too close to next June’s Democratic primary here — and before a new president potentially replaces them in January. While Democratic mayors in this overwhelmingly Democratic city rarely face competitive primaries, this mayor already has four declared challengers with more coming.  Ranked choice primary voting, a system Adams never liked even as he eked out a win in its debut cycle in 2021, means his alternatives won’t necessarily split the change vote.  That’s a tough spot for a mayor who hit an historically low 28% approval rating late last year. The public knew about one federal investigation aimed at City Hall then. Now there are four.  Adams says that he’s done nothing wrong, that voters don’t much care about this, and that he intends to remain in office and win a second term no matter what prosecutors do.  In the meantime, he’s been furiously working to change the subject and narrate his own story.  But as he’s stumping and signifying about “getting stuff done,” the smoke keeps getting thicker. On Monday, two FDNY chiefs were charged with allegedly pocketing nearly $200,000 while taking advantage of a list that started in the de Blasio years to move big-money donors to the front of the line. And three-star NYPD chief Raul Pintos put in his resignation papers, a couple weeks after the FBI seized his devices along with those of then-Commissioner Eddie Caban, who’s since resigned under pressure from Adams, and Deputy Mayor for Public Safety Phil Banks, who’s still on the job with the mayor’s support.  Banks — who abruptly resigned as a NYPD chief in 2014 before reports emerged he’d been an unindicted co-conspirator in a case involving two guys later found guilty of bribing cops and the last mayor — has said mostly through his private attorney that he did nothing wrong then or now.  The deputy mayor — whose former security firm landed a $154 million city contract this year, the New York Times reported on Friday — still hasn’t answered questions about the FBI raiding his home earlier this month.  When my colleague Katie Honan did her job as a reporter and asked him about that on Thursday, he responded with school-boy taunts.  Also on Thursday, NBC News 4 reported that the feds had subpoenaed Our Lady of Mount Carmel Church in Williamsburg.  They were reportedly looking into business dealings between Msgr. Jamie Gigantiello — he was last in the news for letting a provocative music video get shot inside his church — and Frank Carone.  That’s the mayor’s former chief of staff who now runs a lucrative consulting business. His spokesperson recently told NY1 that Carone doesn’t lobby his former boss but combines “his legal, business and media expertise to help clients exceed their goals.” On Friday, the city’s coordinator for asylum seeker services was subpoenaed, reportedly to testify before a grand jury about her meetings with another ex-cop who Adams brought into his administration. That’s Tim Pearson, who runs a shadowy mayoral oversight agency where he allegedly talked about the “crumbs” he expected from city contracts.  That’s according to lawsuits claiming he tried to pressure a female police officer assigned to his agency into sex, then ruined her career along with those of male supervisors who protected her from him. Pearson also had his phones seized by the FBI this month, while taxpayers are covering the bill for his defense in four civil suits at Adams’ behest. Also on Friday, Politico New York reported that City Sheriff Anthony Miranda — yet another ex-cop and old friend of Adams — is being probed by the city’s Department of Investigation for unspecified matters of “public corruption.” That’s a terrible week for Adams, but people writing his political obituary now are out over their skis.  The mayor — a gifted politician who’s been dancing between raindrops since he was a gadfly cop — still has a narrow path to victory next year.  His win and Kathryn Garcia’s second-place finish in 2021 showed that centrist and even conservative ideas appeal to many Democratic voters in a citywide contest, even as his execution has disappointed some of them.  The smoke will clear soon and we’ll see what fires the feds have found — and where that leaves Adams and whoever else ends up in the race.  Siegel ([email protected]) is an editor at The City, a host of the FAQ NYC podcast and a columnist for the Daily News.