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PORTLAND, Ore. — Gov. John Kasich was helping his wife carry some clothes to the basement of their home in Ohio on Wednesday when they broached the subject that many Republicans have been wondering about.

Should he continue his presidential campaign?

After all, Mr. Kasich has not won a single state aside from Ohio. He has fewer delegates than Senator Marco Rubio of Florida, who ended his own bid more than a month ago. And on Tuesday, Mr. Kasich even lost to Donald J. Trump in McKees Rocks, the Pennsylvania community where he grew up.

“I said, ‘What do you think, sweetie? I’m inclined to keep going,’ ” Mr. Kasich recounted at a town-hall-style event on Thursday.

By Mr. Kasich’s telling, his wife, Karen, agreed.

“She said: ‘The people need a choice. And if you don’t give them a choice, who will?’” he told the crowd, which broke out in applause.

“So I’ve decided to keep going,” Mr. Kasich said. “And there are going to be people who are going to criticize me for that. And it’s not always an easy road. I’m going to do my very best.”

Mr. Kasich was counting on the primaries in moderate states in the Northeast to provide friendly terrain for his campaign. But he ended up being no match for Mr. Trump, who won a landslide victory in New York this month and dominated the five states that voted on Tuesday.

Mr. Kasich said that after those most recent contests, he considered the future of his campaign. “I thought about, should I keep going?” he said. “Should I carry on? What is this all about? And I thought deeply about it.”

Speaking to voters, Mr. Kasich said he did not think that Mr. Trump would win enough delegates before the Republican convention this summer to secure the nomination, though he acknowledged it was possible.

“If Donald Trump goes to a convention short of the exact number he needs,” Mr. Kasich said, “he’s not going to get picked.”

Oregon is one of the three states in a fragile, fraying deal between Mr. Kasich and Senator Ted Cruz of Texas, who are desperate to keep Mr. Trump from continuing to pile up delegates. Under the deal, Mr. Kasich agreed not to compete in Indiana, which votes on Tuesday, while Mr. Cruz agreed to bypass later contests in Oregon and New Mexico.

Oregon, which awards 28 delegates, conducts its voting through the mail. Ballots began going out to voters on Wednesday, and they must be returned by the evening of Primary Day, May 17.

Mr. Kasich’s campaign has already rolled out television commercials targeting voters in Oregon. “Ted Cruz pulled out of Oregon,” they declare, “and John Kasich is the only one that can stop Trump and defeat Hillary Clinton in the fall.”

But Mr. Kasich has limited resources for advertising; at the start of the month, his campaign had less than $1.2 million on hand.

And his campaign made an unforced error in Oregon. The official voters’ pamphlet mailed to about 1.8 million Oregon households includes statements from the campaigns of Mr. Trump and Mr. Cruz, along with photographs of the two men, but not one from Mr. Kasich’s campaign.

Oregon’s secretary of state, Jeanne P. Atkins, said the Kasich campaign missed the deadline to submit a statement.