A crushing blow to their knockout hopes was dealt by the fact that they were outclassed by slick Fiorentina in a European match.
The Italians were already two goals up by the time Lewis Neilson was sent off in the second half.
Luka Jovic made up for several misses with a late third to give Fiorentina their first win of the group.
The odds are against Hearts progressing beyond their first European group-stage campaign in 18 years with trips to Florence and Istanbul Basaksehir to come.
One win in 10 matches gave Hearts reason to believe that they could win the European title.
Mandragora was able to plant a simplehead beyond Craig Gordon from Aleksa Terzic's cross in the very first minute.
It was a sign of control. In the first half, Hearts sat deep and invited pressure that came in 71 percent.
The Italians gave the hearts a run for their money. The player who cost Real Madrid £50 million three years ago could have had a hat-trick in the first half hour, but his drive was tipped wide.
The hosts were chasing shadows and just as they were about to make it to the break one down, Kouame acrobatically fired past Gordon after the defence failed to clear.
Josh Ginnelly was denied by Pierluigi Gollini's flailing leg, but that was the only hope left.
Within minutes of the restart, the comeback task was made even harder by the fact that Neilson tugged Jovic's shirt as he broke through.
Gordon made a number of great saves, but the third finally arrived when Terzic found the net after being set up by Kouame.
Hearts were too timid and gave themselves a fighting chance, even though they were clearly inferior. The Italians did not match Basaksehir's four goals.
The manager's tactics will come under scrutiny after they didn't get close to Fiorentina. Lawrence was completely isolated in the first half.
They fought with 10 men than they did with 11. It was a bad night for them and captain Gordon, who has been with the club for two decades.
"We're disappointed with the result, but you have to be proud of the players and the club for getting to this level."
I was disappointed that the goals were preventable. The first two goals are just balls into the box that we should deal with, because they had all the possession and movement of the ball.
The Scottish Premiership game between Hearts and Kilmarnock will be played on Sunday.
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