SEATTLE — In January, Teresa Fuentes’s employer told her she could work no more than 30 hours a week preparing burgers and passing out food at the drive-through for about $9.50 an hour.

It was just the latest blow to her already-fragile household finances. Ms. Fuentes, 47, had missed work to take care of an eye infection for which she could not afford the prescription drops. She is struggling to take care of her two children, reliant on a local food bank, working two jobs and behind on her bills.

“It’s very stressful for me,” Ms. Fuentes said. “Things are just so tight.”

For Ms. Fuentes and thousands of other workers like her, some relief may be on the way. The Seattle City Council is intensely debating a plan to raise the minimum wage to $15 an hour from $9.32 — forging ahead on its plan to tackle income inequality as efforts in the nation’s capital have languished.