The Headingley side have lost all six games following their financial travails in the summer and their points deficit is now 197 after mid-table Hartpury made it three wins from four with a seven-try victory.
Carnegie player-coach, Joe Ford, said: “It’s obviously quite tough at the moment. Although the performances are not yet what we need, the lads are fighting and we’re working hard to get to where we want to be.”
Carnegie’s Tim Bitirim returned at full-back and, among the other changes, wing Tom Varndell and Jamie Elliot also returned after injury, which ruled out Harlequins loanee George Hammond and Sione Faletau.
Hartpury director of rugby John Barnes said after following up a 44-17 victory over London Scottish: “We were pleased to get the win, even though the performance level was slightly down from the previous week.
“We were a bit stop-start, and Yorkshire battled hard, but eventually we got the result we had been targeting.”
Carnegie defended stoutly in the opening 10 minutes and, after No 8 Seb Nagle-Taylor had proved unstoppable from the base of a scrum, they immediately drew level.
They forced an attacking platform close to the line, drove from the base of the scrum and James Flynn went between the posts after several pick and goes.
Hartpury took control and loose-head Dan Murphy and flanker Jack Preece burrowed over to make it 21-7 at the break.
Ryan Shaw was yellow-carded for a high tackle and Hartpury were awarded a penalty try before former Rotherham Titans player Nagle-Taylor added his second.
Joe Batley and Jake Henry also touched down before Carnegie drove and former England winger Varndell peeled off to score.
Hartpury: Simpson, Henry, Koster, Eves, Morris, Bragman, Carter; Murphy, Crane, Gibson, Batley, Safe, Graham, Preece, Nagle-Taylor. Replacements: Tanner, Merrey, Flook, Qera, Hinkley, Cochrane, Goatley, Eden.
Yorkshire Carnegie: Bitirim, Lawson, Shaw, Lancaster, Varndell, Ford, Elliot; Flynn, Sowrey, G Smith, Bloodworth, Humfrey, Lloyd, Seabrook, T Smith. Replacements: Musetti, Davidson, Reid, Gough, Brady, Lavin, Findlay, Carlisle.
Referee: L Pearce.
Sheffield Tigers moved up to fourth spot in National Two (North) with a 21-6 victory over Yorkshire rivals Otley at Dore Moor.
Tigers stand-off Mark Ireland and his counterpart Ben Smith traded penalties early in the match as the score ticked over to 6-6.
But centre Jack Roberts touched down in the 34th minute, and Ireland added the conversion as well as a penalty on 47 minutes.
Finally, winger Pete Swatkins raced over in the final minute to cap the win.
Losing 26-0 with 20 minutes left, Wharfedale battled back with tries from Henry Macnab and Joshua Prell, but it wasn’t enough as they slipped to a 26-14 loss at Chester.
Leading with seconds left, Huddersfield had to settle for two bonus points after hosts Hinckley snatched a 36-33 victory.
A penalty try and another from wing forward Lewis Bradley helped Huddersfield stay in touch at half-time at 17-14, before second-row Nick Sharpe and centre Elliot Munnelly touched down to leave the Yorkshire side 26-24 ahead.
Hinckley edged ahead with a try on 72 minutes, but Bradley’s second try on 75 minutes was converted by Hinchliffe as Huddersfield went 33-29 ahead.
But a last-minute try and conversion saw the hosts take a dramatic win.
And Scunthorpe also lost on the road, Loughborough Students proving too strong and running out 57-11 winners.
Harrogate ran in four tries during a 24-16 victory at Rossendale in the North Premier league, while Sandal edged to a 14-12 victory at Morpeth.
And in North One East, leaders Scarborough suffered a 15-9 defeat at Morley. That meant that York closed the gap with a 24-13 four-try victory over Moortown.