A couple of months before the Month 13 deadline, he got lucky. The Palestinian-Canadian friend who was advising the group called in a favor the sponsors never could have: He phoned an Egyptian immigrant he had helped years ago who now managed a shawarma restaurant in a mall food court and asked if he could give Mr. Hajj a job. Soon, Mr. Hajj was working a couple of evenings a week in addition to continuing English classes.

But when Liz Stark accompanied Mr. Hajj to the bank to deposit a paycheck, she scanned his transactions and noticed something alarming: a couple of thousands of dollars were missing, withdrawn from A.T.M.s. “I thought he was getting scammed, defrauded,” she said later.

Mr. Hajj, who seemed evasive, said he had taken out the money to stock up on food. “I wanted to make up for my kids the nights we used to go to sleep hungry,” he said.

Ms. Stark, believing that Mr. Hajj didn’t understand his accounts well enough to realize what had happened, went to the bank to try to figure out what had gone wrong. When that turned up no evidence of theft, she and the other sponsors wondered if there were other explanations for the unfamiliar pattern. Had Mr. Hajj sent the money to his father in Syria? Stashed it away in a drawer?

A few weeks later, Mr. Hajj asked the sponsors about going on welfare. He had heard about it from his classmates in English lessons. Some were enrolling, seeing it as a safer bet than insecure, low-wage jobs, they told him. One explained that he could work and still collect the government assistance, if he could persuade his boss to pay him under the table.

With a sinking feeling, the sponsors began to worry: Had Mr. Hajj been withdrawing the money to game the system, to lower his bank balance so he would qualify for social assistance?

Even legal use of the welfare system by Syrian refugees was a charged question, the sponsors knew. Some Canadians argued that welfare was a necessary step for some refugees, buying them more time to learn the language, which would lead to better jobs in the long term. A survival job could turn into a trap, the philosophy went.