The work by Western Power Distribution (WPD) is to replace 460m of low voltage cable that has come to the end of its working life, and is set to be done by June 28.
Urgent replacement of the cable is required to maintain and improve security of supply to all customers connected to it.
During the works there are temporary traffic signals and periods of phased road closures on Phoenix Bank, High Street, Church Street, Shropshire Street and St Mary’s Street.
A WPD spokesman said: “WPD is continuing with emergency works in Market Drayton town centre and has almost completed the installation of the new mains electricity cables.
"Works to transfer the 68 properties that will be serviced by these cables will be completed by June 28.
"WPD appreciates your patience while these emergency works are completed.
“We would like to apologise for any inconvenience caused to residents and motorists.”
Shropshire Council will continue to review and monitor the works as they progress.