For President Obama, how those competing priorities balance out could mean the difference between securing a landmark accomplishment — the first overhaul of the nation’s immigration laws since 1986 — or becoming consumed by charges of scandal.

Invigorated by the uproars, House Republicans are setting their sights more firmly this week on the I.R.S. and Mr. Obama’s embattled attorney general. After weeks of trying to leaven the House’s growing investigatory zeal with serious legislating, House leaders and committee chairmen appear to be giving themselves over to an expanding and aggressive oversight effort — on the I.R.S., the Justice Department’s targeting of reporters, Attorney General Eric H. Holder Jr.’s statements to Congress on that targeting and the Sept. 11 attack on the United States Mission in Benghazi, Libya.

House leaders including Speaker John A. Boehner of Ohio have acknowledged the risk if voters see the investigations as driven primarily by politics. But with the legislative season moving toward the routine task of passing spending bills, oversight appears to be the biggest splash that the House hopes to make…

The White House plans to keep the president in the public eye, concentrating on kitchen-table issues like the economy and the carrying out of Mr. Obama’s health care law as well as on high-profile foreign-policy efforts. The hope, officials say, is to prevent Congress from seizing a public agenda that has largely been set by Mr. Obama this year.