While there has been a rapid pick up in vaccinations - more than 90 per cent of the population have now had at least one dose, rates among the more vulnerable elderly have lagged the rapid spread of the Omicron variant in the city.
Only about 53 per cent of those over 80 years old are vaccinated, according to government data.
Lam said that her government was shifting its approach to target the elderly with more medical resources in its drive to combat the virus after a senior Chinese official said that Hong Kong needed to prioritise reducing deaths.
The government had previously focused on identifying, treating and isolating each infection even if they were mild or asymptomatic cases. It was also planning to hold a mass testing scheme for the city's 7.4 million residents in March.
Lam this week said that there was no time frame for the testing and it would only be done if there were sufficient resources.
Hospitals, morgues and isolation centres are overflowing while some supermarkets and grocery stores still have shelves empty nearly two weeks after residents started panic buying in fear over a potential city-wide lockdown.
Lam said that she was the "authoritative source" for all coronavirus information and would address any rumours or misinformation. Many residents have been frustrated and anxious at what they see about constant changes to the government's policy.
Lam said that her administration was managing the outbreak as best as possible with the policy changes made in order to achieve "the objectives of safety and health".
"This is an unprecedented situation, in terms of responding to an epidemic of this magnitude, it has outgrown the capacity of the Hong Kong government," she said.