Many Labour candidates in Birmingham cited the bin strike as the reason people were telling them they would not be voting for the party while canvassing.
Speaking to the BBC on Friday, an emotional Cotton said he would have liked to have solved the year-long dispute sooner, but there were “complicated issues” including around equal pay for men and women.
“I’m sorry we weren’t able to find a way to do that earlier,” he said.
However, Majid Mahmood, the council’s cabinet member for environment and transport, managed to hold on to his seat in Bromford & Hodge Hill.
As the council’s cabinet member responsible for waste services, he has faced a turbulent 16 months.
Cotton added the day was “not a time for personal sadness”, saying: “This has to be about jobs and growth and the future of our communities and whoever comes next, I hope they focus on that.”
Asked about Prime Minister Sir Keir Starmer and if he could keep his job, Cotton said: “I think the Labour party’s challenges are wider than one particular office holder or individual.
“It’s clear we haven’t communicated our message and our story clearly enough to the voters.”