Financial records reviewed by The New York Times show that Mr. Cohen, President Trump’s personal lawyer and longtime fixer, used the shell company, Essential Consultants L.L.C., for an array of business activities that went far beyond what was publicly known. Transactions adding up to at least $4.4 million flowed through Essential Consultants starting shortly before Mr. Trump was elected president and continuing to this January, the records show.

Among the previously unreported transactions were payments last year of about $500,000 from Columbus Nova, an investment firm in New York whose biggest client is a company controlled by Viktor Vekselberg, the Russian oligarch. A lawyer for Columbus Nova, in a statement on Tuesday, described the money as a consulting fee that had nothing to do with Mr. Vekselberg.