WASHINGTON—House Democrats are grappling with whether to pursue further investigations of President Trump following his acquittal in the Senate, facing both an election in nine months and fresh White House actions that they say demand scrutiny.

Democrats want to look into whether the president improperly influenced the Justice Department’s sentencing recommendation for a Trump confidant, casting Mr. Trump as emboldened by the end of the impeachment process. At the same time, party leaders are eager to focus on pocketbook issues...