The phrase "next breath" refers to the following moment when someone inhales air into their lungs. It is used to emphasize the immediate future or how quickly something can happen. Full definition
But in the very next breath, each religion claims it's My favorite one. (religion.blogs.cnn.com)
The majority leader said that there was money for mass - transit capital improvements, and then in the very next breath said there was also money for bridges and highways, which, he said, are in a state of disrepair. (politico.com)
It's interesting that in one breath Mr. Cabreras can claim that the government can't answer the questions of how people should deal with the tragedy, and in the very next breath claim that they are obligated to do exactly that by giving him or someone else like him the forum to proclaim their one particular take on it that excludes other options. (religion.blogs.cnn.com)