Kent 343 (Khushi 53, Barker 4-85, Leaning 118) and 132 for 2 (Compton 71*) are 30 runs behind Hampshire 505 for 8 December.
Kent had a chance to salvage a Vitality County Championship draw with Hampshire thanks to goals from Jack Leaning and Ben Compton.
After returning from injury to score 118 in the first innings, Leaning led the charge. In the second innings, Compton chipped in with 71 not out.
Nevertheless, Hampshire had to follow on after falling behind by 162 runs in the first innings, and they had just eight wickets remaining at the end of the match, which tested their resolve.
With the score down to only 30 runs, the pair finished the day with an 84-run partnership with Kent, bringing them up to 132 for 2.
Day three was never more than sedate, following day one’s daring 211 by James Vince and day two’s hectic schedule. After leading the visitors through an uncomfortable evening session in which they ended up six down and trailing by 299 runs, Leaning made his comeback on 73.
Although Vince dropped him at first slip on 84, he managed to recover and showed remarkable leadership by gathering the tail.
Leaning sustained a broken hand during the tie with Worcestershire in May, which caused him to miss the previous seven weeks of action.
He amassed an undefeated 179 in that innings, and in only 207 balls, he achieved back-to-back hundreds. This marks his tenth career century, his sixth since joining Kent, and his second at Utilita Bowl in his past three trips.
Two wickets went on either side of the second new ball around Leaning.
Two overs before the new ball was revealed, Charlie Stobo’s reasonable twenty-eight was cut short when James Fuller pinned him leg before. And then Mohammed Abbas slapped Grant Stewart on the pads four balls into the new cherry.
However, Matt Parkinson reduced the margin with a routine 24 as he spent more time out of the game with Leaning. In an attempt to annoy the hosts, the two scored 44.
But when Leaning’s magnificent knock came to an end due to a splayed stump by Kyle Abbott, Parkinson bowled Kent out for 343 after Barker finished with 4 for 85.
Almost avoiding the follow-on, the final four Kent wickets put up 137, but Vince was able to turn them around.
Although Feroze Khushi was more subdued than in his exhilarating 53rd over, he was taken out by a short-ball trap that allowed Barker to top edge him to long leg.
Daniel Bell-Drummond held on for a little while longer, but Liam Dawson eventually gained enough momentum to cannon into an off-stump. However, Compton became friends with that man Leaning, who helped to ease the pain of the day.
Opener Compton reached fifty in 114 balls with a reverse sweep.
Leaning was dropped by Vince for the second time in the day but was otherwise solid to all but wipe out the deficit.
Hampshire will need eight wickets and likely a bit of a chase on the final day, and might be without Barker – who pulled up during a delivery midway through the evening and didn’t return.