MANCHESTER, N.H.—Former New York City Mayor Michael Bloomberg has devoted millions of dollars of his personal wealth to campaigns to curb gun violence and address climate change. He has often cast his fortune in virtuous terms when it comes to politics, noting the degree of independence it gave him while he served a dozen years as mayor.

“Americans want elected officials to put partisanship aside and work together on real solutions to their problems because they see these problems all around them,” Mr. Bloomberg said Tuesday...