Clarke is keen to land the full-time post at Sincil Bank, but his chances took a knock as Bournemouth stepped into a two-goal lead inside 20 minutes. Brett Pitman opened the scoring on seven minutes in spectacular fashion when he volleyed home Steve Fletcher's knockdown from 12 yards.
The Cherries doubled their lead on 20 minutes, when Lincoln's defence failed to deal with Liam Feeney's long throw and Sammy Igoe's half-hit shot bobbled past an unsighted Rob Burch.
Lincoln got themselves a lifeline on 25 minutes with their first shot at goal when Chris Fagan controlled the ball with his back to goal and spun to volley spectacularly into the top corner.
But any hopes of a Lincoln fightback were dashed when Bournemouth made it 3-1 on the hour. Feeney broke from the halfway line and exchanged passes with Marvin Bartley before crossing for Igoe to spectacularly volley his second.
Pitman spurned two further chances to extend Bournemouth's lead and, in the final minute, Lincoln's misery was complete when Aaron Brown's free-kick struck the angle of post and bar.