WASHINGTON—Mike Johnson’s grace period is likely over with House Republicans, with griping and obstructionism firing up again from different wings of his party, undercutting his efforts to unify the GOP heading into a rebooted budget fight.

After winning the gavel in late October, the Louisiana Republican averted a crisis this past week and passed a short-term spending bill that funds the government through early next year. But to do so, the new speaker had to rely on large numbers of Democrats, with 95 members of his own party voting against that deal because they said it did too little to cut federal spending or secure the border.