CLARENCE, N.Y. — In early June, Gov. Andrew M. Cuomo joined forces with Nancy Pelosi, the House Democratic leader, and declared in no uncertain terms that he would do whatever he could to help unseat six Republican members of New York’s congressional delegation in next year’s midterm elections.

Since then, more than a dozen Democrats have announced their candidacy in those six districts, according to documents filed with the Federal Election Commission, joining nearly a dozen others who had preceded the governor’s declaration of war.

The Republicans are certainly taking the threats to their seats seriously, especially considering the historic trend of losses for the president’s party in midterm elections. And there is no question that the stumbles of the Trump administration and the still-stinging November loss by Hillary Clinton have encouraged a groundswell of energy in the Democratic Party base.

Yet the decision by Mr. Cuomo to wade into the federal races has added an interesting dynamic to the contests — one that the Republicans do not seem to mind at all.