“These debates have offered our only guaranteed opportunity to share my vision with the American people,” his campaign wrote in an email to supporters, saying the money would fund outreach in early-voting states to get Mr. Castro the polling results he needs to qualify. “If I can’t make the next debate stage, we cannot sustain a campaign that can make it to Iowa in February.”

A similar gambit worked for Senator Cory Booker of New Jersey, who raised $1.7 million in the last 10 days of September after saying he would drop out otherwise.

Biden will take super PAC help after all

Joseph R. Biden Jr.’s campaign dropped its longstanding opposition to receiving assistance from super PACs on Thursday, opening the door for wealthy supporters to spend unlimited amounts of money to try to lift him in the Democratic primary.

Mr. Biden had explicitly renounced super PAC support for his 2020 run, so the move was a stark reversal and an implicit acknowledgment of his weakened position. He entered October with only $9 million in the bank, far behind his leading rivals.

Faiz Shakir , the campaign manager for Senator Bernie Sanders of Vermont, immediately criticized the flip. And Senator Elizabeth Warren of Massachusetts tweeted that it was “disappointing that any Democratic candidate would reverse course and endorse the use of unlimited contributions from the wealthy to run against fellow Democrats.”

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Who’s winning? The polls don’t know

Four polls out this week presented a muddled picture of how the Democratic primary is unfolding.