Museum supporters say that discord accompanying the firing may distract from efforts to support poor and working artists from the Latin diaspora and to raise the institution’s international stature. The new director, Patrick Charpenel, a respected curator in Mexico City, will be under pressure to steady an organization with deep roots in New York City but one that has cycled through three executive directors over the last seven years and has reduced its hours of operation.

The fired executive, Berta Colón, who had served as the deputy director of institutional advancement, said she was told on May 19 at the museum that she was being dismissed for “performance reasons.” But she disputed that assertion in a letter to the board of trustees, writing that the museum would face a deficit of more than $800,000 by the end of the fiscal year and that anticipated revenue numbers she had been provided were “not based on solid expectations and in some cases were grossly inflated.”

She went on to accuse Carlos Gálvez, who was named with her in August to lead the museum temporarily, of employee intimidation. “Staff is threatened with the possibility of being fired, they are pitted against each other,” Ms. Colón wrote. “During this period of transition without an executive director, Carlos has created an environment that promotes distrust, fear of retaliation and isolation.”

Mr. Gálvez did not return calls to a phone number in his name seeking comment for this article, and a spokeswoman for El Museo said in an email that “the museum is declining to comment at this time.” Questions that remain unanswered include whether the museum is looking into the accuracy of Ms. Colón’s assertions and whether the incoming director, Mr. Charpenel, was involved in the decision to fire her.

The controversy comes as the museum, on Fifth Avenue’s Museum Mile, awaits the arrival of Mr. Charpenel, who was born in Mexico and was named as director on May 1 but has not yet obtained the paperwork needed to work in the United States, according to two people close to museum officials. The two spoke on condition of anonymity because they did not have permission to discuss the matter publicly.

El Museo was formed in 1969 by artists and activists to emphasize Puerto Rican cultural contributions but eventually embraced a broader mission: to exhibit the work of Latino, Caribbean and Latin American artists from all backgrounds. Its permanent collection of over 6,500 objects includes pre-Columbian Taíno artifacts and 20th- and 21st-century drawings, paintings and sculptures. Among other tasks, Mr. Charpenel will have to balance sometimes competing views of how to preserve the museum’s activist roots while furthering its global ambitions. He will also have to expand fund-raising, which may require skills that he didn’t need at his previous job, heading a privately financed museum, Museo Jumex, in Mexico.

And he will have to navigate the bureaucracy of the City of New York, which owns the museum’s building, on Fifth Avenue and 104th Street, where a tall scaffold has stood for years in front of the facade as part of planned repairs.

Now he may also have to deal with the fallout from the letter by Ms. Colón, who wrote that the way she handled her duties had never been formally evaluated before her firing. Ms. Colón added that she and other directors and staff members had recently discussed Mr. Gálvez with a museum trustee.

In her letter to the board, which was obtained by The New York Times, Ms. Colón wrote that during what was supposed to have been a confidential vetting process, Mr. Gálvez met with museum workers, conveyed information about the four leading candidates for the executive director slot and identified two he thought the staffers should support, suggesting jobs could be lost if other candidates were selected.

Yasmin Ramirez, an ex officio museum trustee appointed by the de Blasio administration, referred specific questions about Ms. Colón’s allegations to the board chairwoman, Maria Eugenia Maury, but also offered her own thoughts.

“I applaud the choice of Patrick because of his history of supporting and exhibiting collectives and artists involved in social practice work and who articulate critical perspectives on art and politics and the human condition,” Ms. Ramirez said by phone. “At the same time I am troubled by the recent dismissal of Berta and what I read in her letter about employee intimidation, and that has to be investigated at Museo del Barrio, especially given our history of supporting artists and audiences from poor and working classes.”

Since a highly regarded director, Julián Zugazagoitia, left El Museo in 2010, financial problems have forced staff cuts, and the museum went from being open six days a week to five. Mr. Zugazagoitia’s successor, Margarita Aguilar, was fired after 18 months, then filed a claim of gender discrimination and workplace hostility against the museum’s board that was ultimately rejected by the New York State Division of Human Rights. The next director, Jorge Daniel Veneciano, departed unexpectedly after two years.

Mr. Charpenel was trained as a philosopher and is described by El Museo as a curator and collector. His most visible post was as the director of the Museo Jumex (pronounced WHO-mex) in Mexico City, which opened in 2013 as a showcase for the private collection of Eugenio López Alonso, the sole heir to the Grupo Jumex juice fortune, and is run by a foundation financed by that company.

Although El Museo’s programming still includes shows, like one in 2015 on the history of the political group the Young Lords, that fit with its original mission, some supporters and scholars saw the appointment of Mr. Charpenel as emblematic of a move further away from the barrio the museum is named after. Karen Mary Davalos, a professor of Chicano and Latino Studies at the University of Minnesota, said that the museum’s broadened mandate could serve as a “smoke screen for dilution.”

“How much real estate, how much programming will go to Puerto Rican artists?” she said. “What is the cost to the community whose blood, sweat and tears went into forming El Museo?”