Nigeria reported Tuesday that the leader of Boko Haram and several of his top commanders were killed or wounded in an airstrike, just as Secretary of State John F. Kerry arrived in the West African nation with a message that military action alone won’t break the terrorist outfit’s grip on the region.

“Beating Boko Haram on the battlefield is only the beginning,” said Mr. Kerry, who called on Nigeria to “strike at the root causes of violent extremism” by more aggressively fighting rampant corruption in the government, as well as policies of marginalization and inequality that the terrorists have exploited to recruit followers.

His comments drew loud applause, but it was too early to determine how the message would resonate with Nigerian leaders, who have spent months pushing Washington to ease arms sales restrictions that the Obama administration has kept in place amid concern over corruption and human rights abuses.

The Nigerian military offered few details on what types of planes or bombs were used in the Tuesday strike, but said the air force carried out a “most unprecedented and spectacular air raid” that killed about 300 militants and “fatally wounded” the group’s leader, Abubakar Shekau.

Nigerian officials have claimed on at least three occasions that they have killed or fatally wounded Shekau — only to have him resurface weeks or months later in video recordings. Two statements by the military Tuesday did not give the basis for the latest claim but identified several other of the terrorist group’s commanders as “confirmed dead.”

There was no immediate reaction from Boko Haram, which for years has made headlines with its grisly use of female suicide bombers and its mass kidnappings of schoolgirls since launching an insurgency in northeastern Nigeria in 2009, with a name that loosely translates as “Western education is sin.”

An estimated 15,000 people have been killed by the group, which controls territory the size of Maryland along northeastern Nigeria’s borders with Niger, Chad and Cameroon. U.S. intelligence officials have said Shekau is bent on establishing an Islamic caliphate.

Shekau pledged allegiance last year to the Islamic State, although connections between the groups remain unclear.

Early this month, the Islamic State issued a statement saying it was naming a leader for what it described as its West African operations. Shekau later appeared to contradict that account in a video message of his own.

Nigerian military officials said the air force struck while Shekau was praying Friday at Taye village in the extremists’ Sambisa Forest stronghold in northeastern Nigeria. “Boko Haram terrorist commanders confirmed dead include Abubakar Mubi, Malam Nuhu and Malam Hamman, amongst others. While their leader, so-called ‘Abubakar Shekau,’ is believed to be fatally wounded on his shoulders,” said a statement by Col. Sani Kukasheka Usman, an army spokesman.

Three months ago, the International Crisis Group argued that Boko Haram had been weakened and “put on the defensive” by a growing and coordinated military response by Nigeria and its neighbors.

The Brussels-based think tank said that under Nigerian President Muhammadu Buhari, who came to power last year, “Nigeria has regrouped, and neighbors are collaborating with it more meaningfully.”

Officials from Cameroon, Chad, Niger, France, Britain and the U.S. gathered in Nigeria in May to discuss the campaign, which the Obama administration has backed with roughly $200 million in humanitarian assistance for Boko Haram-affected populations in the region.

President Obama said in October that 300 U.S. troops would be deployed to a secretive base in Cameroon, but administration officials have been loath to speak publicly about the extent of Washington’s involvement in military aspects of the campaign.

Nigerian officials have been pushing for Washington to sell them aircraft to expand the fight. But such sales were halted during the presidency of Goodluck Jonathan amid corruption concerns and reports that Nigerian troops had carried out flagrant rights abuses against captured insurgents and innocents deemed to be backing Boko Haram.

Mr. Buhari argues that Nigeria’s human rights record has improved enough to lift the arms blockade. In May, U.S. officials told Reuters that Washington wanted to sell up to 12 A-29 Super Tucano light attack aircraft to Nigeria in recognition of Mr. Buhari’s reform of the army. Congress still needs to approve the deal.

Mr. Kerry made no reference to the agreement Tuesday but praised the Buhari government for making “a terrific start — a strong start at all levels of government” by “working with civil society to encourage official transparency and accountability.”

“It is so important to restoring trust among the people,” Mr. Kerry said before holding talks with Mr. Buhari. “His administration has taken important steps to prevent the theft of public funds and to recover stolen assets.”

But the secretary of state suggested that far more needs to be done. “Nigeria is already a regional leader in many ways — economically and culturally — but you can also become a model in fighting corruption and the organized crime that so often goes along with it,” he said.

“To effectively counter violent extremism, we have to ensure that military action is coupled with a reinforced commitment to values like integrity, good governance, education, compassion, security and respect for human rights,” Mr. Kerry said. “Values that the terrorists don’t just ignore, my friends, but values that they desecrate and try to destroy at every turn that they can.”

• This article is based in part on wire service reports.

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