But it is American officials who are responsible for the stalling of one batch of gear. For weeks, officials in Washington have debated whether to accept a donation of masks from the Chinese Foreign Ministry, American officials said. Though medical workers are desperate for the masks, some officials argue that taking the donation would aid China’s propaganda campaign.

That debate is a sign of the growing fury and frustration in Washington over that campaign. American officials who see the Chinese Communist Party as malevolent are irate at what they view as the party’s efforts to reshape the narrative of the pandemic, which spread quickly beyond central China in part because of cover-ups by officials. (The State Department did not reply to questions on the offer or its position on donations from China.)

The American officials say China is trying to turn the spotlight away from the pandemic’s roots through the shipments of purchased supplies and some donations. Often, Chinese officials tell counterparts abroad that they must publicly thank China in return for the shipments, say Western officials, executives and analysts with knowledge of the exchanges.

“What is most striking to me is the extent to which the Chinese government appears to be demanding public displays of gratitude from other countries; this is certainly not in the tradition of the best humanitarian relief efforts,” said Elizabeth C. Economy, the director for Asia studies at the Council on Foreign Relations. “It seems strange to expect signed declarations of thanks from other countries in the midst of the crisis.”

The equipment has helped save lives, and officials around the world — among them, Gov. Andrew M. Cuomo of New York — have thanked the Chinese government or private citizens.

Chinese state-run news organizations have highlighted the exports in stories and social media posts.

Some Chinese officials abroad have been surprisingly aggressive in pushing Beijing’s narrative.

Twice since late February, a Chinese diplomat in the consulate in Chicago has emailed State Senator Roger Roth, the president of the Wisconsin Senate, to ask that that Senate pass a resolution recognizing that China has taken heroic steps to fight the virus, according to reviews of the emails by The New York Times.