They were a goal down after seven minutes, and a man down after 26.
Much of the blame for Saturday's two-goal defeat was landed at Nathan Eccleston's feet, after his challenge on Gills' Stuart Lewis earned him a straight red.
Clark labelled the challenge as "reckless" but at least it fired up battle-weary Huddersfield.
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Goalkeeper Alex Smithies had been called upon twice to deny Rene Howe before strike-partner Simeon Jackson struck.
Town best chance in the first half came when Garry Richards almost put Anthony Pilkington's cross into his own net.
Just 20 seconds into the second half and Jackson scored again after bursting clear.
Substitute Gary Roberts hit the post during a frantic finale but they were unable to get past League One's best home defence and slipped out of the play-offs.
Clark said: "We didn't respond from Tuesday night and we still had a hangover from that. We were deservedly one down but when the sending off happened we responded positively.
"We will certainly discipline Eccleston because his lack of discipline possibly cost the side. It was a reckless challenge. The players showed great spirit but it was a massive task.
"We need to dust ourselves down and not to sulk and be big babies."
Gillingham: Julian; Lewis, Richards, Dennehy, Nutter; Miller, Weston, Jackman, Barcham; Jackson, Howe (Oli, 73). Unused substitutes: Royce, Maher, Bentley, Palmer, Dickson, Walker.
Huddersfield Town: Smithies; T Clarke, Trotman, P Clarke, Williams (Robinson, 74); Pilkington (Roberts, 61), Kay (Collins, 54), Drinkwater, Eccleston; Novak, Rhodes. Unused substitutes: Eastwood, Berrett, Pearce, N Clarke.
Referee: F Graham (Essex)
Man of the match: Alex Smithies.
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