South Korean President Park Geun-hye replaced her prime minister and two other top officials Wednesday in a bid to contain a political crisis, as accusations of influence-peddling and other activities prompted prosecutors to request an arrest warrant for one her closest personal friends.

Opposition lawmakers were quick to dismiss the Cabinet reshuffle and called for Ms. Park’s resignation, but analysts said she is likely to survive and suggested that the scandal wouldn’t impact major policies such as Seoul’s increasingly hard line toward North Korea.

There is no question, however, that the final year in office for South Korea’s first female president will be tumultuous, filled with public backlash as polls put her approval rating at about 10 percent — the lowest since Ms. Park’s inauguration in 2013.

Wednesday’s developments followed a swirl of eye-opening twists in a scandal that has plagued Ms. Park since she acknowledged last week that her longtime friend Choi Soon-sil had edited some of her speeches and provided public relations help despite having no official government position.

Ms. Park made the acknowledgment during a surprise public apology a day after the South Korea TV network JTBC reported that it had obtained a discarded personal computer of Ms. Choi’s containing some 44 files that included drafts of some of Ms. Park’s speeches and Cabinet meeting remarks.

South Korean media have had a field day since, speculating that Ms. Choi, the daughter of a onetime religious cult leader, had access to sensitive government information and likely played a much larger role in government affairs, including in shaping Ms. Park’s position toward Pyongyang.

With regard to criminal implications, news outlets have reported accusations that Ms. Choi has pulled government strings from the shadows over the past several years — even pushing businesses to donate millions of dollars to two foundations she controlled to obtain money for her personal use and for Ms. Park’s postretirement activities.

The scandal gained steam Monday, when Ms. Choi appeared in public to surrender to authorities. What followed was a melee-type scene, in which hundreds of pushing and shoving reporters and protesters descended on the small 60-year-old woman outside the Seoul prosecutor’s office.

At one point, police reportedly stopped a protester from confronting Ms. Choi with a bucket of animal feces. She lost her Prada shoes after being knocked to the ground several times.

Social media went wild as images of the abandoned black footwear and the word “Soonderella” — a play on Ms. Choi’s name and the fairy-tale heroine who left behind her glass slipper — spread internationally.

On Wednesday, prosecutors submitted an official request to the Seoul Central District Court to approve an arrest warrant for Ms. Choi, and a court spokesman said authorities likely will determine by Friday whether to approve it. Ms. Choi will remain in custody until then.

Prospectors also summoned Ahn Jong-beom, a former secretary to Ms. Park, for questioning Wednesday related to his suspected involvement in extracting $70 million in potentially illicit donations from companies tied to the Choi accusations.

North Korea angle

Ms. Choi is a daughter of the late Choi Tae-min, the leader of a pseudo-Christian and shamanistic cult believed to have served as Ms. Park’s professional mentor after her mother was assassinated in 1974.

Ms. Park’s father, former South Korean President Park Chung-hee, was gunned down by his intelligence chief in 1979.

In acknowledging her ties to Ms. Choi last week, Ms. Park said the woman had helped her “when I had difficulties.”

Analysts say one of the big unanswered questions is whether Ms. Choi truly influenced Ms. Park’s state affairs.

The full scope of documents that may have been transferred between Ms. Park and Ms. Choi is not clear.

South Korean media have reported that documents sent to Ms. Choi included information on secret military talks with North Korea and that she received daily official reports and made policy suggestions to Ms. Park.

“The major factor here that could influence South Korea’s policy toward North Korea going forward would be if it is discovered that Choi’s fingerprints were on the policy,” said Scott Snyder, who heads the program on U.S.-Korea policy at the Council on Foreign Relations in Washington.

Should that be the case, it may suggest that Ms. Park’s push during recent years for a more aggressive posture toward Pyongyang resulted from an undemocratic process lacking transparency, Mr. Snyder said.

“The policy would come under scrutiny in terms of its legitimacy, and I think there are people in the opposition who will grab onto that and make it an issue,” he said. “But right now, we don’t have that yet.”

Mr. Snyder also downplayed the prospect that Ms. Park’s Cabinet shakeup would have any serious impact on the direction of her presidency. The prime minister’s position in South Korea is largely administrative. Ms. Park also announced nominations to replace her financial services commission chairman and minister of public safety.

History of scandals

Political strife is far from new in Seoul.

An analysis published this week by the Center for Strategic International Studies in Washington outlined how similar scandals have — in one way or another — plagued every one of the nation’s presidents dating back to the late 1980s.

Written by Northeast Asia analysts Victor Cha and Lisa Collins, the CSIS analysis notes how Ms. Park’s predecessor, Lee Myung-bak, was mired in revelations that his elder brother and close aides had accepted bribes from two Korean banks.

Mr. Lee’s predecessor, Roh Moo-hyun, was nearly impeached during his 2003-2008 term amid accusations that he had illegally interfered in South Korea’s legislative elections, a scandal that was later seen to drive Mr. Roh to suicide in 2009.

Nobel Peace laureate Kim Dae-Jung, who held the presidency from 1998 to 2003, was enveloped in political crisis involving his sons, who faced charges in 2002 of exploiting his office for money and real estate.

The analysis also notes that former Presidents Roh Tae-woo and Chun Doo-hwan were convicted in 1996 of corruption, mutiny and treason for their roles in a 1980 massacre and the 1979 assassination of Ms. Park’s father, although both were later pardoned.

“Politics is undeniably a blood sport in Korea, and it looks as though the embattled Park will lose a pint or two as well,” Mr. Cha and Ms. Collins wrote.

But they predicted that the South Korean president is unlikely to be driven from office, saying “at least at this point, such an outcome is unlikely.”

Mr. Snyder suggested that it is a wait-and-see situation. He also recalled a political poll from years ago that asked South Koreans what quality they would most want in a president.

“At the time, many responded that a president should have no sons, because that would be the only way to avoid corruption,” Mr. Snyder said. “Well, Park has no sons, but she has friends. So, now it seems the only way to avoid political corruption in South Korea is for the president to have no sons and no friends.”

• This article is based in part on wire service reports.

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