Harriet Dart knows she needs to improve at clutch moments after a nightmare first-set tiebreak in her second-round defeat against Marta Kostyuk at the US Open.

Britain’s Dart, who had already served for the opener twice, squandered five set points against the Ukrainian 19th seed – three of them on her serve.

Kostyuk had already missed three of her own in a marathon tiebreak but took her fourth opportunity and then ran away with it to seal a 7-6 (10) 6-1 victory in a match which saw the temperature hit 38 degrees Celsius.

“I served for the first set three times. Had a lot of set points, it happens,” he said. “I just also was very hot.

“Credit to her. She played well at the important times. And of course, I’m disappointed to not take my chances.

“Sometimes you didn’t even think just because you’re so hot and a bit bothered.

“I feel like anyone I step on court against now, I’m confident I can take it to them.

“It’s just few matches recently, at some important moments, situations have happened and that’s something I need to work on and get better at and just get better at closing those.

“Whether it’s matches, sets, certain situations, but the main thing is to be to be there competing with the very best.”

Defeat means Dart could not reach the third round of a second successive grand slam following her run at Wimbledon in July, but she is set for a new career high ranking, provisionally in the high 60s.

It also means her teacher mum Susie, who has been her stand-in coach in New York, will not have a scheduling dilemma as she is due to return on Friday to prepare for next week’s new term.

Susie has been in the limelight over the last 48 hours and has enjoyed the exposure.

“My dad found it very amusing that the dog got mentioned a lot,” Dart added.

“I think she just found it quite amusing. I think she had a few texts from some school parents saying, ‘Have you left? Have you taken a new job?’.

“But she’ll be going back to school. So she’s definitely enjoyed her time.”