NHS England believes a total of 5,261 people have been affected.
Steve Russell, NHS national director for vaccinations and screening, said: "This issue has now been fixed and everyone affected will be offered support and any catch-up screening as soon as possible, including where they may now be above screening age.
"We would like to extend our sincere apologies to those affected for this error and any additional worry this may have caused.
"Anyone who is concerned they may have missed an invitation to screening can call our dedicated helpline for support and further information."
Around 15 million people are invited to NHS screening every year.
NHS England says it has issued guidance to GP Practices and Integrated Care Boards to make sure all patient registrations are fully completed, and has put in place new measures to ensure all eligible people are invited for screening.
Anyone in this group that is no longer eligible for screening will be able to have their screening if they choose to opt in.
Details of how to do this are set out in the letters that have been sent to those affected.