When it comes to airing dirty laundry in Washington, dry-cleaning abuse has to be near the top of the list.Secretary of State Mike Pompeo, under fire for demanding that his agency’s chief watchdog be dismissed, is just the latest in a long line of capital power players accused of dispatching aides to the corner cleaner because they were too impossibly busy to do such mundane chores on their own.Time after time, administration officials and lawmakers who run afoul of the ethics rules are found to have sent their subordinates to drop off and pick up their freshly laundered shirts and blouses. The ethics troubles that led in 2018 to the downfall of Scott Pruitt, the former administrator of the Environmental Protection Agency, included claims of dry-cleaning runs by his security detail. Multiple members of Congress have also gotten rung up on dry cleaning charges.“It is always the dry cleaning,” said Lisa Gilbert, a top official at the advocacy group Public Citizen. “It always seems to start there.”While the offense itself might seem to be a trivial and peculiar kind of misdeed in a city known for its strict business-attire dress code and a surfeit of dry cleaners, veteran ethics enforcers in Washington say such improper use of staff often represents a willingness to blur the line, meaning a deeper look can find more egregious acts.Dry-cleaning malfeasance can be a gateway drug.“Once you are doing that, what’s to stop you from asking people to do more errands or other things that might be bending the rules?” said Noah Bookbinder, the executive director of Citizens for Responsibility and Ethics in Washington. “In the long tradition in Washington of people misusing official resources to benefit themselves personally, this is sort of an easy first step.”In Mr. Pompeo’s case, it is unclear what role the alleged dry-cleaning misconduct may have played in any investigations that drew his ire, ultimately leading him to call for the ouster of the State Department’s inspector general. The watchdog, Steve A. Linick, was looking into Mr. Pompeo’s handling of an $8 billion arms sale to Saudi Arabia, a significant foreign policy matter. But Democratic aides have also said that Mr. Linick was scrutinizing Mr. Pompeo’s use of agency staff for his personal convenience, such as walking his dog and fetching his dry cleaning.Good-government advocates say that directing publicly paid staff members to do everyday chores reflects a sense of privilege and self-importance and a disregard for public service norms, which hold that aides are supposed to advance the public interest, not provide for an official’s personal comfort.“They have a sense of entitlement that people are there to serve them, not necessarily the United States government,” said Fred Wertheimer, the president of Democracy 21 and a longtime advocate of government accountability in Washington. “If you don’t have a sense of what the role is of the people who work for you and what the role is not, then you are going to have this occur.”One of the reasons the handling of dry cleaning gets so much ethics attention is that it is easily understandable and can serve as shorthand for a variety of unofficial requests or demands that are prohibited by ethics rules, such as taking care of pets, running to the store and shuttling around family members.But the sheer frequency with which it has turned up as a problem is remarkable.The year after Mr. Pruitt resigned under an ethics cloud, a spokeswoman for the Pentagon, Dana W. White, was found by the inspector general there to have misused her staff for personal errands such as — you guessed it — retrieving dry cleaning. Ms. White, who had already resigned, blamed disgruntled civil servants for the complaints and said the assignments were related to their duties.Dry cleaning regularly figures into ethics cases on Capitol Hill as well. Representative John Conyers Jr., the longtime Democratic member from Michigan ultimately forced out in 2017 over a sexual harassment complaint, had been accused in 2006 of improperly requiring staff to perform tasks such as retrieving his dry cleaning and doing other household chores. The case was resolved at the time after Mr. Conyers acknowledged a “lack of clarity” in his communications with staff members regarding their official duties and responsibilities.In 2011, the Citizens for Responsibility and Ethics in Washington filed a complaint against Representative Laura Richardson, Democrat of California, based on internal emails that showed she required staff members to attend fund-raisers and, of course, pick up dry cleaning as part of their jobs. She was reprimanded by the House in August 2012 for improper use of staff and lost her re-election bid that year.In 2018, Politico published an article about Representative Tom Garrett, Republican of Virginia, that described how Mr. Garrett turned his staff into virtual personal servants. Though the piece somehow didn’t mention dry cleaning, it did note that staff members were ordered to bring him clean clothes from his apartment. Mr. Garrett chose not to seek re-election and said he would seek treatment for alcohol issues.Over the years, other cases have popped up of lawmakers and officials in the executive branch essentially enlisting staff members as personal assistants when they are on the public payroll.Such situations can present a quandary for aides, who are dependent on the officials for their jobs and know that any griping can lead to unemployment. Some staff members are more than happy to do menial chores as they climb the Washington ladder, regarding them as just another set of tasks in service of their boss’s success. But that doesn’t make it right, and agreeing to extracurricular duties often leads to more such requests.Leaders of the watchdog groups say they believe the current problem may be aggravated by the culture of the Trump administration, in which many top officials have little experience in government service but plenty in the private sector, where they are accustomed to being coddled by assistants.“We saw a corporate cabinet,” Ms. Gilbert said. “And what that means is we see folks who are not necessarily putting the mandates and the mission of their agency first, but are thinking of making a profit and personal services.”Indeed, President Trump did not seem too troubled on Monday about the accusations that Mr. Pompeo might have been looking for help with minor chores when other family members were not around to assist.“They’re bothered because he’s having somebody walk his dog, as you’re telling me?” Mr. Trump said when questioned by reporters. The president suggested it might be fine for Mr. Pompeo to have asked for aid if he were otherwise occupied “negotiating with Kim Jong-un — OK? — about nuclear weapons.”“I don’t think it sounds, like, that important,” he said, adding that the inquiry suggested to him that the nation’s “priorities are really screwed up.”Mr. Trump’s thoughts aside, the history of government ethics cases points to a common theme when officials order aides to see to the laundry: In the end, it is usually the taxpayers who are being taken to the cleaners.