Most of the customers affected are in Scotland, with the possibility that some will have to wait until Wednesday for repairs.
Scottish and Southern Electricity Networks (SSEN) said that as of 12pm on Tuesday, power has been restored to 105,000 customers with around 9,500 customers remaining off supply.
A spokesperson for Northern Powergrid said: "Our teams are still working out in the field to complete the restoration of supplies to the remaining customers affected by Storm Malik.
"We have 61 properties that we are working to reconnect as soon as possible today.
"Our teams got through a phenomenal amount of work yesterday, which included reconnecting around 17,000 customers that were impacted by Storm Corrie.
"Almost all of the work we have left to do is a blend of repair work that we discovered when we got to site, some difficult site conditions that meant we couldn't complete the work yesterday or where we would have needed access to the property during the night but agreed with the customer to complete the restoration today."
SSEN said the main areas which continue to be affected are rural Aberdeenshire and the Angus border, with some customers also off supply in Perthshire, the Highlands, Western Isles and the Moray coast.
A spokesperson said: "Further good progress is being made today, and SSEN expects to restore power to the vast majority of customers by this evening, prioritising areas of the network where customers have been without power the longest.
"For small pockets of customers, this may extend into Wednesday - depending on the extent of damage found and the impact of weather on our teams' ability to work safely."
Trees which had stood for centuries were snapped in two by hurricane-force winds that raged across the north of Scotland on Sunday night, bringing down power lines, shutting roads and crushing graveyard head stones.