A Head Start program in Western Massachusetts started its school year Monday, two weeks later than usual to save money due to sequestration-related budget cuts.

On Tuesday,the program almost shut down again.

“We would have had to say, ‘We are glad you came today. I’m sorry, you can’t come tomorrow,’” said Clare Higgins, executive director of the Community Action of Franklin, Hampshire, and North Quabbin Regions, which provides Head Start services to more than 500 children in a 1,700-square-mile-region.

The organization has an annual contract with the federal government that renews each Oct. 1. But with the government shutdown, the grant was not renewed this year.

The program was spared its own shutdown because the state stepped in and provided enough funding to keep it going for two more weeks, Higgins said.

Around the country, 23 programs in 11 states did not get the annual grant they had been scheduled to receive Tuesday. The National Head Start Association estimated that without intervention, the shutdown could cause as many as 19,000 children to lose Head Start services.

The interruption in funding follows sequestration cuts that led to the elimination of services for 57,000 children and their families.

“This kind of instability for kids and families and staff is not good,” Higgins said.