• Public schools, which often buoy the September jobs report as districts rehire teachers, bus drivers and administrators, proved a drag on last month's report.
• The September 2021 jobs report shows employment decreased by 144,000 in local education and by about17,000 in state education on a seasonally adjusted basis.
• That shortfall may be related to "difficulty in hiring for certain positions such as bus drivers, food service workers, or substitute teachers," says economist Elise Gould.

Public schools often buoy the September jobs report as districts around the country hire teachers, bus drivers, administrators and other staff back for the start of the fall term.

That did not happen as it normally does in 2021.

Government employment, which includes public education at both the state and local levels, was a notable drag on the September 2021 jobs report with a loss of 123,000 jobs in the public sector on a seasonally adjusted basis.

Breaking down that number further, employment decreased by 144,000 in local government education and by roughly 17,000 in state government education on a seasonally adjusted basis. That means that while hiring at public schools did increase, the bump was not large enough by historical standards to offset the seasonal adjustment and resulted in a net loss.

"The bounce back in local government education was weaker than expected for schools reopening in September and seasonally adjusted K-12 employment fell by 144,000," Elise Gould, an economist at the Economic Policy Institute, wrote in a Twitter post after the report's release.

"The shortfall may be related to recent reports of difficulty in hiring for certain positions such as bus drivers, food service workers, or substitute teachers, often lower-wage and part-time positions filled by many older workers who may be more concerned about infection," she added.

Employment at public schools remains challenged despite the rollout of the Covid-19 vaccines this year, with local school districts still down more than 300,000 jobs since February 2020.

"Most back-to-school hiring typically occurs in September. Hiring this September was lower than usual, resulting in a decline after seasonal adjustment," the Labor Department said in a release. "Recent employment changes are challenging to interpret, as pandemic-related staffing fluctuations in public and private education have distorted the normal seasonal hiring and layoff patterns."

The anemic public sector print weighed on the headline September 2021 jobs report, which showed the U.S. economy added just 194,000 jobs last month. That was well short of the 500,000 expected. The unemployment rate fell from 5.2% to 4.8%.

CNBC studied the net changes by industry for September jobs based on data contained in the government's employment report.

While economists were quick to blame the unseasonable education figures for a disappointing payrolls number, most industries in the private sector posted decent numbers last month.

The leisure and hospitality sector, which stalled in August with little change in hiring, rebounded with a net gain of 74,000, with continued job growth in arts, entertainment and recreation that combined accounted for 43,000.

Notably, hiring at restaurants and bars held steady for the second consecutive month, compared with an average monthly gain of 197,000 from January through July. Employment in leisure and hospitality is down by 1.6 million, or 9.4%, since February 2020.

"With a small increase of 74,000 jobs in September after a weaker August — again likely the result of the surge in the Delta variant at the end of the summer — leisure and hospitality has a jobs shortfall of nearly 1.6 million since February 2020. We are still in a pandemic!" Gould added.

The wide-ranging professional and business services sector added 60,000 jobs in September. Hiring continued to increase in architectural and engineering services, management and technical consulting services, and computer systems design and related services.

— CNBC's Nate Rattner contributed reporting.

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