"Public Sector Jobs?"
You mean like Teachers, Fireman, and Police?
Yup.
Where have Republicans advocated cutting these jobs?
Teachers, Fireman, and Police are almost all LOCAL government employees (outside of Annapolis, the AF Academy, and West Point, for example, there are no Federally employed teachers). Their pay is based on LOCAL budgets, which are LOCALLY funded.
The reason Local funding isn't enough for additional teachers is because Federally Mandated regulations require school districts to spend LOTS on things OTHER THAN TEACHERS (e.g. a nurse to change Johnny's diaper so he will have an Equal Opportunity to succeed with other 11th graders).