Pollock, South Africa's main strike bowler and a powerful middle-order batsman, misses the match to attend the birth of his first child, with the two sides locked together at 1-1 in the series following England's 70-run victory in a fluctuating third test at Trent Bridge on Monday.
"We will miss Shaun, he is a world-class cricketer but it is up to others to take responsibility," Smith said.
"Gary Kirsten should be fit to return to the side and we are confident we can continue to play the fantastic cricket we have done to dominate the series until this match."
Kirsten, the most experienced batsman in the South African squad, missed the third Test because of an arm injury but should be fit to return at Headingley, possibly in place of Boeta Dippenaar, who failed twice at Trent Bridge.
Seam bowler Dewald Pretorius will also probably come in, though he will have to perform well to fill the boots of Pollock, who took six wickets in England's second innings with his nagging fast-medium deliveries to bring his career total of test victims to 299.
Vaughan believes Pollock, who also contributed a handy 62 with the bat, will be a loss, though he still expects South Africa to bounce back, just as his side did from their drubbing in the second test at Lord's.
"He's one hell of a player to replace," Vaughan said.
"Shaun is a world-class performer, especially on wickets which suit his style of bowling, But the pressure is now on us to continue where we left off at Trent Bridge and we know South Africa will come back at us hard."
Vaughan said England had made a candid assessment of their poor performance in the second Test.
"We were honest after Lord's and said we didn't play well as individuals or as a unit," he said.
"Headingley should be a result wicket and if we exploit the conditions we will have a chance."
England are likely to make one change to their side because fast bowler Steve Harmison picked up at calf injury at Trent Bridge, with experienced Surrey seamer Martin Bicknell and Worcestershire fast bowler Kabir Ali added to the squad.
"It was important to win at Trent Bridge and we all feel elated but we realise it's only 1-1," Vaughan said.
"There was a bit of tension and a few nerves early but we stuck to our disciplines well."
Smith is confident his team can recover their momentum quickly.
"We are not too down on ourselves, just a little irritated to lose the Test match. We did well to take the match to five days and I think lesser teams might have crumbled," he said.
"Day one of a Test match is always important" Smith added.
"The first morning sets the tone and it's about who performs best on the day. We will go to Headingley and practise hard. Now it's our turn to bounce back."