Several schools in Aberdeenshire said they will be shut or delay opening on Monday as they struggled with power and heating problems.
Storm Corrie moved eastwards across Scotland on Sunday and is set to push across the North Sea in the early hours of Monday.
Winds of 92mph were recorded in Stornoway, on the Western Isles, as Storm Corrie began to hit the UK on Sunday night.
Scotland’s First Minister Nicola Sturgeon said the impact of Storm Corrie is “likely to be significant”.
She tweeted on Sunday: “Work to repair the damage from Storm Malik continues. Tens of thousands have had power reconnected already – however, many will remain off supply again tonight and some, especially in north east, could be off into Tuesday. Welfare arrangements are in place.
“Special arrangements remain in place for vulnerable customers and local resilience partnerships continue to work with councils to provide welfare support.”
Strong winds are expected until the early hours of Monday and they may affect parts of Norfolk, where large waves could hit the coast.
Weather warnings for wind across northern parts of Scotland are in force.
Forecasters said that people should beware of flying debris that could lead to injuries, and there may be some damage to buildings including tiles being blown from roofs.
Mr Burkill said: “It is not just the case of strong winds causing problems – there is also the ice risk across parts of Scotland through to the early part of Monday morning.
“There will be some wintry showers. Emergency services are trying to get out, utility companies are trying to make repairs and so the icy conditions are not going to make that easier for them.”
Ice warnings were issued by the Met Office covering Grampian, Highlands and Eilean Siar, Strathclyde and Fife on Monday as falling temperatures after Storm Corrie may see some snowfall and turn untreated surfaces icy.
On Sunday evening 7,500 households were thought to be without power by the end of Sunday, the Scottish Government said in an update at 7pm on Sunday.
Deputy First Minister John Swinney said: “The power companies have drafted in a large number of additional engineers and are making significant inroads into reconnecting customers.
“However, we need to be aware that the arrival of Storm Corrie could hamper these efforts and add further problems.
“For those who will unfortunately not have power tonight, support with alternative accommodation is available to anyone who needs it.”
Richard Gough, of Scottish and Southern Electricity Networks (SSEN), said “we expect the full restoration of customer supplies from both storms to extend into the early part of next week”.
Rural Aberdeenshire is among the hardest-hit areas and some customers in Angus, the Highlands, the Moray coast and Perthshire were still waiting for supplies to be restored.
ScotRail had withdrawn services from 6pm and in an update this morning said “After the storm, Network Rail will have to complete safety checks on some routes before we can open them back up for passenger services. Some of these routes are already complete but some can only be done in daylight.”