Wales' First Minister has revealed the NHS recently experienced its busiest ever day since the health service was founded.
He said there are still pressures across the system as a result of a "very large number" of people waiting to be discharged.
It comes following advice from the Welsh Government to the health boards which suggested some patients may be sent home from hospital without care packages to cope with the demand and to free up bed space in hospitals.
Mr Drakeford said that on 27 December, 550 people were admitted to hospital across Wales on that day alone.
"The 27 December was probably the busiest day the NHS has ever experienced in its 70-year history", he told the press conference.
He apologised to the people who have had to "wait a long time to access care".
The Welsh Government later added that pressures on that day were reported "across the urgent and emergency care system".
“NHS management information shows there were particularly high levels of 111 calls, 999 red ambulance incidents and Emergency Department attendances. This was likely due to a mix of seasonal respiratory conditions, as well as coronavirus, influenza and concern among parents for children regarding Strep A. ”
During the press conference, Mr Drakeford added the week ahead will continue to be "challenging" as ambulance workers are set to stage further walkouts this month.
Mr Drakeford said he would not be revealing the detail after announcing healthcare workers in Wales will be offered a one-off payment as part of a bid to avoid more strikes.
Over the weekend, he said the offer was part of a package of measures the Welsh Government had put to trade unions in a letter on Friday.
Doctors in Wales are also considering industrial action and members of the Royal College of Nursing (RCN) have already taken part in two days of strikes in December.
When asked if he was embarrassed at the current state of the health service he said he "wished things were better."
"I wish people didn’t have to wait the lengths of time they do and I wish the physical conditions under which staff work and people sometimes have to wait could be better than they have been."
"But I do think it is important and fair to step back for a minute... [latest figures] show waits are down at every point in the system that we are seeing more people in 21 weeks from the start to the finish of their treatment in any time since the pandemic.
"We have 20% fewer emergency admissions in the Welsh NHS than we did in the winter before the pandemic because of all the extra things we have put in place.
"A record number - going on for 15,000 in a single month - learned they did not have cancer because the system had carried out tests.
"So while the system is under huge pressure , and I'm keen to emphasise none of this is to minimise that, the system goes on managing to see hundreds of thousands of people every month and to do that successfully. "