Left-back Smith, 30, had been playing with a hip injury and will undergo an operation on Monday.
And forward Josh Magennis is also missing with an ankle knock, with Clark expecting him to be out for at least 10 days.
"We've lost two crucial players," said Clark shortly before Saturday's lunchtime match against Celtic.
"But there's been lots of players since I've come here saying they're worth a go so now they have the opportunity.
"The 18 we've picked today is the 18 players we have fit and available in the football club."
Clark hopes Northern Ireland international Magennis, who was called up for their friendlies against Wales and Slovenia, will be back early next month.
The 25-year-old is the top scorer at Rugby Park this season with 11 goals.
"Josh has jarred his ankle in training, late on Tuesday evening," Clark told BBC Scotland.
"He came in Wednesday after I had done the media, ready to train but didn't feel it was right. We scanned him. We've informed Northern Ireland he'll be out for a minimum of 10 days so we'll hopefully have him back for Partick Thistle [on 2 April].
"Huge blow, because he's got power, pace. His confidence was up after playing so well in the second half at Aberdeen [last week's 2-1 defeat], he got a good goal.
"So, obviously it's disappointing but just as more so for me we've lost Steven Smith also. We've lost him for the season. He now has hip surgery on Monday.
"He was outstanding at Aberdeen and I was confident that we could we just manage him through but it's proved too much.
"It must be too much because for him to even make himself available at Aberdeen shows the commitment he's got."
Kilmarnock manager Lee Clark has confirmed defender Steven Smith will be out for the rest of the season.