England boss Gareth Southgate was in the crowd at the DW Stadium to run the rule over Manchester City loanee Hart, who was England's first-choice stopper during the World Cup qualifiers but has since lost his West Ham place.
The 30-year-old has not played a Premier League game since November and will be used in no further FA Cup matches by David Moyes after his side lost 2-0 at Wigan to exit the competition.
Hart could do little about either of Will Grigg's goals - the first was a header powered into the top corner from close range and the second was a penalty slotted down the middle - but the keeper conceded there was a sense of personal disappointment with Southgate watching on.
"Everything that could have gone wrong went wrong but I think we've only got ourselves to blame with a number of the incidents that have gone on, so it's very disappointing," Hart told the club's website.
"We conceded early but I don't think that's a good enough excuse. You go 1-0 down, you pick yourself up and get back in the game. It's a cup game and you keep fighting.
"It goes without saying that we didn't take our chances to impress. It's very disappointing on my behalf, because I want to perform and losing 2-0 at Wigan is difficult, so I'm struggling for positives."