A second-half header from Peter Whittingham saw Cardiff City take the spoils in south Wales and left the Baggies teetering on the brink of the drop zone.
West Brom produced little in the way of chances in what turned into a dogged midfield battle between two sides fighting to get clear of the relegation zone.
Cardiff struck just after the hour and Clarke concedes the Hawthorns club are going through tough times as the busy festive period approaches.
"It's not a good moment for the team, it's not a good moment for the club," he told Sky Sports.
"Nobody is enjoying it, we are trying as hard as we can to try and turn the corner.
"Today was one of those games where the first goal was always going to be vital and unfortunately it went to Cardiff."
A fourth straight defeat leaves West Brom 16th, just two points clear of the bottom three, and Clarke believes it is time to face reality if the Baggies are going to retain their top flight status.
"The performances are OK but sometimes you have to separate yourself from the performances and look at the results and that's what we have to do now," he added.
"The stark reality is that we have lost four matches in a row and we have slipped down the table.
"We are looking over our shoulder which is never a comfortable thing to do.
"We have to work as hard as we can to stop this run of defeats as soon as possible, starting next week at home to Hull City."
Clarke's men take on Hull at The Hawthorns next Saturday before facing a testing trip to White Hart Lane to face Tottenham on Boxing Day.