In New Mexico, Representative Michelle Lujan Grisham, prevailed in the Democratic primary for governor. If elected, she would be the first Latina Democratic governor in the United States, replacing Gov. Susana Martinez, a term-limited Republican who is also Latina. Steve Pearce, a conservative Republican congressman, won the party’s nomination.
The race to replace Ms. Lujan Grisham in the House included Deb Haaland, who was campaigning to be the first Native American woman in Congress. Ms. Haaland, a former state Democratic Party chairwoman, has far outpaced her rivals in fund-raising for a seat that is expected to remain blue.
Among the other states voting on Tuesday were South Dakota, where nominees for governor were being chosen, and Mississippi, which held its primaries for a Senate seat occupied by Roger Wicker, a Republican who won his primary Tuesday night and is expected to be re-elected without major incident.
But Democratic leaders have trained their attention most intensely on to Republican-held House seats in blue states like New Jersey.
In the Seventh District along the western part of New Jersey, Representative Leonard Lance easily defeated two challengers Tuesday to win the Republican primary, and is expected to face his first tough general election in a decade for a seat that has been held by Republicans since 1981. His opponent will be Tom Malinowski, a former assistant secretary of state, who had raised more than $1.2 million in his effort to capture the Democratic nomination.
Part of the intense focus on seats in New Jersey involves the unpopularity of Mr. Trump and his policies. The administration’s early signature accomplishments, most notably the overhaul of the tax system, took aim at high-tax states like New Jersey. And the plan to open offshore drilling off the Atlantic Coast drew widespread condemnation up and down the Jersey Shore, a cherished resource and tourism destination.