It comes as NHS Lothian asked staff to volunteer for additional shifts to fill gaps at the Royal Hospital for Children and Young People and elsewhere across the service.
A combination of Covid cases, staff self-isolating and annual leave has left many health boards struggling, with NHS Lanarkshire warning demand in its hospitals is as high as any point during the pandemic.
The number of people estimated to have Covid by the Office for National Statistics has risen – from one in 90 to one in 80, the highest level since the infection survey began in October.
Some 1,505 new cases of Covid were reported on Friday, with a 6.4 positivity rate.
The number of people in hospital has fallen, to 502, while 57 people are in intensive care.
And Scottish Labour warned the third wave has hit Scotland’s care homes, as cases have tripled.
Janis Butler, Director of Human Resources and Organisational Development at NHS Lothian, said the health board has asked staff to pick up additional voluntary shifts to cover the shortfall.
"We are currently experiencing staffing pressures across our system due to the number of staff who are off work because they are self- isolating, either as a contact of a Covid case or because they have Covid themselves, as well as the effects of the summer annual leave period,” she said.
“In response to this, we have offered staff in some of our services the option to cover additional shifts if they choose.
"This is just one of the mitigating measures we take to carefully managed and maintain safe staffing levels with the required mix of skills and experience and ensure the wellbeing of our very hardworking staff.”
North and South Lanarkshire HSCPs said services are “extremely stretched”, and asked family and friends to step in and take on some at home caring responsibilities in non-urgent cases.
NHS Lanarkshire is prioritising urgent services for the most vulnerable, it said, as “sustained pressure” across the service continues.
Hospitals in the area are as busy as they have been during the pandemic, said Judith Park, director of acute services.
“Covid numbers in our hospitals are rising and this is an additional pressure while we are trying to recover services and treat patients who have planned operations,” she said.
“Pressures on our staff, and those in our two partnerships, is also a concern and they are struggling to cope with the sharp rise in demand.”
The issue is due to an increase in Covid cases, as well as staff shortages due to self-isolation, attempts to recover services after lockdowns, an increase in complex cases in the system, and a “massive increase” in A&E visits, the health board said.
Care at Home and Home Support services are the worst affected.
Marianne Hayward, interim chief officer, South Lanarkshire Health & Social Care Partnership, said: “The pressures being experienced across the whole health and social care system are exceptional.
“The current circumstances have forced us to move to critical service delivery across all areas. This means we are prioritising our services for the most vulnerable.
“This will result in discussions with service users and their families/carers about reductions in care packages where it is safe to do so and identifying support so that the most vulnerable people continue to receive critical services."
Ross McGuffie, chief officer, Health & Social Care North Lanarkshire, said: “This is the most challenging point for staff delivering safe, effective healthcare across the whole of health and social care since the pandemic started."
He added: “We have recruited more than 100 new staff across the North partnership, however the level of pressure the whole health and social care system is under means that we’re still facing major challenges in how we deliver services.
“Care at Home/Home Support services are now extremely stretched across North and South Lanarkshire.”
Scottish Labour warned of a “growing crisis” in care homes, as the number of homes with Covid cases tripled in two months.
There are now 59 adult care homes with a proven or suspected case of Covid-19, up from 18 two months ago, according to figures from Public Health Scotland.
Scottish Labour Covid Recovery spokesperson Jackie Baillie said: “The third wave of Covid has now undeniably reached our care homes.
“The way care home residents have been failed time and time again during this pandemic is nothing short of a national scandal. We cannot let history repeat itself.”
The most recent figures from the Office for National Statistics show an increase in cases in Scotland, though the ONS said the increase was not entirely clear.
Prof Kevin McConway, Emeritus Professor of Applied Statistics at The Open University, said this was “not too much comfort” in light of the recent easing of restrictions which is likely to lead to more infection.
He said: “In England, the numbers of people who would test positive for an infection, in the most recent week, is about where it was in late January this year.
“In Wales and in Northern Ireland, it’s relatively quite a bit lower, but still roughly at the level from mid-February.
"In Scotland, despite the fact that the trend isn’t very clear in the most recent week, the estimated number of infected people is at the highest level it has been since the CIS began there in October last year.