Mr Sunak promised “hard-headed common sense” in approaching the migration issue and blamed the pandemic for creating huge backlogs in the health service.
“Our country faces some of the biggest challenges any of us will have seen in our lifetimes, with our economy, in our NHS, on our borders,” Mr Sunak said in the clip, first broadcast on ITV.
“Covid has done huge harm to the NHS.
“It’s created massive backlogs, which we’re working through, but Britain’s heroic response to Covid shows that where the political will exists to really grip a problem, we can bring everyone together and achieve the unthinkable.
“That takes leadership, so I’ve immediately set in train a new approach and taken the best of our experience from fighting Covid to reduce NHS backlogs and get patients treated quicker.
“I’m determined to get the number of people on waiting lists falling and I will not let you down.”
The Prime Minister admitted earlier on Wednesday that he had used private health care, following weeks of speculation over whether he relied on NHS services.
He said he was registered with an NHS GP but told MPs he had used “independent” health care in the past.
In the video, the Prime Minister also touted the “difficult but fair decisions” he has made to get borrowing and debt under control.
Rishi Sunak through the years — in pictures
On addressing illegal migration and small boat crossings in the Channel, Mr Sunak said: “We’re taking decisive action already, deploying hundreds of new agents to tackle immigration criminals, putting an end to the appalling situation where taxpayers are paying to keep illegal migrants in hotels.
“And we’re introducing new laws that make it unambiguously clear that if you come to our country illegally, you will not have the right to stay and will be removed.
“Britain will always be a place for global talent and those in dire need.
“But basic human decency must be accompanied by hard-headed common sense.”