Electricity companies are hoping to have reconnected every home that lost power during Storm Otto on Friday by the end of the day., with about 2,000 in Aberdeenshire still without power.
Meanwhile, the Met Office said the storm has "well and truly cleared" and that Otto, which left more than 60,000 homes without power, has moved on to the continent and is now affecting Scandinavia.
Gusts of 75mph to 80mph were recorded across parts of northern Scotland on Friday while trains and flights were cancelled and roads were blocked by overturned lorries in northern England.
Scottish and Southern Electricity Networks (SSEN) said it had restored power to around 41,000 properties since the storm struck.
It has sent food vans to the main areas still cut off from supply and has been serving food and drink since Saturday morning.
A yellow warning for snow and ice had been in place for central parts of Scotland overnight but milder conditions are now expected over the weekend.
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In England, Northern Powergrid said around 21,000 customers lost power on Friday, but just one property was still affected as of 8.30am on Saturday.
A spokesperson for the company said: "It was a fantastic effort by our teams to restore power to 21,595 customers as a result of the storm, across what was a challenging day."
However, the weather will be mild across southern and central areas of the UK. Scotland is expected to see sunny spells further south in the afternoon.
On Friday, a man was taken to hospital in a serious condition after a tree fell on a street in Sheffield.
A South Yorkshire Police spokesperson said: "A man in his 50s was injured and was taken to hospital in serious condition. A property nearby was also damaged and structural engineers are at the scene."
Elsewhere, a tree toppled on to a Porsche on Granby Road in Harrogate, North Yorkshire.
Met Office meteorologist Craig Snell said Saturday would remain "breezy", particularly along the west coast, but nothing "on the scale that we have had".
Temperatures are expected to reach 14C (57.2F) to 16C (60.8F) in Herefordshire on Saturday and sunny spells are expected in southern England.
Mr Snell said: "For many of us, for the time of year, it is not a bad February day.
"Tomorrow across England and Wales it is going to be a fairly decent day if you like bright and mild weather, with a lot of sunshine.
"Scotland and Northern Ireland are a bit cloudier, wetter and windier, with a risk of gales but not on the scale we saw with Otto.
"On Sunday temperatures will be fairly mild and will reach 14-15C in southern and western England.
"Next week it is set to turn a good deal colder."