WASHINGTON — The Supreme Court is heading into the final month of its annual term amid lingering questions over whether the eight ideologically divided justices will be able to decide politically charged cases, including ones on abortion and immigration.

Since the February death of Justice Antonin Scalia, the justices have struggled at times. They split 4-4 and could not rule on a conservative challenge to the mandatory union fees that are paid by millions of public employees, and they did not rule on whether the Obama administration’s promise of free contraceptive coverage for female employees infringed on the religious liberty of the church groups that run religious nonprofits, including schools and colleges.

Instead the justices issued a short opinion that said both sides in the dispute — the Catholic bishops and the Obama administration — had agreed on a compromise that could resolve the matter.

Here are some of the biggest remaining cases awaiting a decision by the end of June:

Immigration and presidential power

Case: United States v. Texas

Question: Did President Barack Obama overstep his authority as chief executive when he announced the government would defer deportation and offer work permits to about 4 million immigrants here illegally who are the parents of U.S. citizens or legal residents?

Background: Texas and 25 other Republican-led states sued and won orders from two lower courts that put the president's plan on hold. Obama's lawyers argue the president has broad power over how to enforce immigration law. Moreover, they say Texas has suffered no harm, and therefore has no standing to sue. Lawyers for Texas say the president is trying to change the immigration laws in defiance of Congress. During the oral argument in April, the justices sounded evenly split with the conservatives backing Texas and the liberals supporting Obama's plan. A 4-4 tie would deal a defeat to the administration by keeping in place the lower court's order.

State restrictions on abortion clinics

Case: Whole Woman's Health v. Hellerstedt

Question: How far can anti-abortion lawmakers go to restrict abortion clinics through health and safety regulations?

Background: The Texas Legislature in 2013 decided abortions could be performed only at clinics that meet the standards of an outpatient surgical center and employ doctors who have admitting privileges at a nearby hospital. These rules, if put into effect, would close about 30 of the state's 40 abortion providers, said lawyers for the clinics, leaving vast areas of South and West Texas with no licensed abortion provider. Lawyers for Texas say states have broad authority to enforce health and safety standards. In the past, the high court said states may not put an "undue burden" on women seeking an abortion, but has not clarified exactly what that means. Justice Anthony M. Kennedy holds the key vote. He has regularly voted to uphold abortion restrictions, but during the argument, he sounded skeptical of Texas' claims.

Affirmative action and college admissions

Case: Fisher v. University of Texas

Question: May a state university still use racial preferences to complement its admissions process, even if it has been able to enroll significant numbers of Latinos and African-Americans without using affirmative action?

Background: Abigail Fisher, a white student who was rejected by the University of Texas, says she was denied the equal protection of the laws because race was a factor in who was admitted. The university says its use of race is both limited and necessary for achieving diversity on campus. The court's conservatives, including the late Justice Scalia, said they wanted to restrict or abolish race-based decisions. With his death, they lost a key vote. Now the court will rule but with seven justices, not including Elena Kagan because of her early work in the Obama administration. Kennedy probably holds the key vote here as well. He has voted against affirmative action in the past, but did not join Scalia in seeking to abolish it. While a broad ruling could affect universities nationwide, the justices may rule narrowly by focusing on Texas and its so-called "Top 10 percent" plan.

Public corruption and bribes

Case: McDonnell v. United States

Question: Was the former Virginia governor properly convicted of bribery after he took $175,000 in gifts and loans from a promoter of a dietary supplement and then set up meetings with state officials who could have — but did not — undertake research on the benefits of the supplement?

Background: The case of Robert McDonnell could weaken the anti-corruption laws and make it hard to prosecute officials who do small favors in exchange for cash. McDonnell is appealing his bribery conviction by arguing he did not take "official action" in return for money. He said the arranged meetings do not count because he did not order or pressure officials to take action. During the argument, the justices, both liberal and conservative, sounded as though they agreed with McDonnell's claim that bribery charges require proof that an official made a government decision, such as awarding a contract, in exchange for a payoff. Justice Department prosecutors warned this would give a green light to state and local officials to take cash payoffs for opening doors and setting up meetings.

Drunken driving and search warrants

Case: Birchfield v. North Dakota

Question: Do suspected drunken drivers have a right to refuse an alcohol test if a police officer does not have a warrant?

Background: All the states say they will revoke the licenses of drivers who refuse to be tested. Minnesota and 12 other states go further and make it a separate crime to refuse a breath or blood test. Those laws are being challenged by civil libertarians who rely on a 2013 Supreme Court ruling that said under the Fourth Amendment, officers generally need a warrant before forcing a driver to submit to a blood test at a police station. If the high court were to broadly extend that rule to routine breath tests, it could force changes across the nation. The justices during the argument pondered whether police may obtain a warrant by phone in a few minutes and whether a delay could dissipate the alcohol in the driver's blood.

Police stops and exclusionary rule

Case: Utah v. Strieff

Question: Can police stop a pedestrian or motorist without a good reason, see if they have an outstanding warrant, and if so, then search them for evidence? Or does the "exclusionary rule" forbid the use of evidence that was found during such an illegal stop?

Background: A case from Utah may decide how much leeway police officers have to stop and question pedestrians. The court's conservatives have been inclined to shrink or even abolish the "exclusionary rule" because it requires judges to reject evidence that was obtained illegally. For the same reason, liberals insist it is necessary for enforcing the Fourth Amendment's ban on "unreasonable searches." The Utah Supreme Court threw out the drugs found on a Salt Lake City man who was stopped and searched by an officer who discovered he had an outstanding warrant for a traffic offense. Prosecutors said the warrant justified the search, but the state's judges said the illegal stop required suppressing the evidence. During the argument, the liberal justices noted that a surprisingly high percentage of residents in many cities, including Ferguson, Mo., have outstanding warrants for not paying small fines.

Guns and domestic violence

Case: Voisine v. United States

Question: Does a guilty plea for a "reckless" act of misdemeanor domestic violence allow for a lifetime ban on gun ownership?

Background: Twice in recent years, the court has upheld a 1996 federal law that takes away the gun-ownership rights of someone who has been convicted of domestic violence, including a misdemeanor charge. The justices agreed to hear an appeal from a Maine man who was charged with illegal possession of a hunting rifle under federal law because he had on his record an earlier guilty plea in a state court to a "reckless" act of domestic violence for slapping a girlfriend. The justices sounded as though they will again uphold the broad reach of the federal law, but over a dissent by Justice Clarence Thomas. During the argument, Thomas broke his decade of silence and asked a government lawyer whether any other misdemeanor conviction could result in the loss of a constitutional right, referring to the Second Amendment right to own a weapon.