The Ayrshire outfit have endured a difficult start to the season, with a run of poor results leaving them second bottom of the Scottish Premiership.
They suffered another setback last weekend in the shape of a 2-0 defeat to Motherwell but Clark says Killie are ready to pick themselves up and go again at Tannadice on Friday night.
He said: "We looked closely at last week's game. The goals were horrendous and we did look vulnerable at other times in the match.
"We are trying to work on that and we have to stop that happening.
"When you look at the attacking play, we actually played a lot of good football.
"There is no reason why we can't go into the game with confidence and know we are capable of doing well.
"The players' spirits are fine.
"There is no point telling lies to anybody. We have got failings within the team and it's mainly in the defensive area.
"We came to the club and we didn't have any centre-backs with experience and the ones we have brought in have been injured or suspended.
"We had Manuel Pascali playing at the back last week and Jackson Irvine has played at the back.
"It's been nothing other than bad luck that we can't get a settled back four and we've still got issues there for tomorrow.
"But that's no excuse. We've got to be more solid as a team to give us a chance of winning games."
The cup competition may provide some respite from league action but Clark knows Kilmarnock could scarcely have asked for a tougher fourth round tie.
He added: "It's a different kind of pressure. You look at the way things have been for us and we've not had a lot of breaks.
"In the Scottish Cup draw, we got another Premier League club who are doing well.
"It's not ideal but that's what happens when you're in the situation we are in. But it's a game where we can go and express ourselves.
"United are a good team and we think we've got good players who are capable of doing well against them.
"It's a cup tie so it's a slightly different focus as in you've got to make sure you get stuck in and do things right and that's what we will certainly be trying to do."
Meanwhile, reports this week suggested Nigerian international Gabriel Reuben has told the club he wants to quit in January.
The midfielder has not featured in the first team this season, with boss Allan Johnston stating last week he had to work harder to win a place in the side.
Clark said: "There's no football issue whatsoever with any of our players and the staff. Gaby is in the same category as everyone else.
"He's had a couple of bad injuries and he played part of the under-20s game the other day. There's no problem there at all.
"If any of our players are unhappy, they can move in January.
"It's as simple as that. But I think you'll find that the vast majority are happy."