The Tigers were backed by a big travelling support as they eased their Premier League relegation worries with an impressive 4-0 win in Cardiff on Saturday.
Hull have been handed an FA Cup replay with Brighton, little more than 48 hours later, and the winners will face Sunderland on their own turf.
Livermore insists the players will shrug off their tiredness and focus on securing a last eight fixture back at the KC Stadium.
"Every game is massive but we've been dangled a great carrot with another home tie after this one," said Livermore.
"It's a massive thing for the club to keep striving for. We want to be a big club and we want to win things.
"We had an amazing amount of fans at Cardiff and we owe them something to look forward to as well as ourselves."
The midfielder netted the fourth goal against Cardiff and is keen to play his part again for the Tigers in their pursuit of silverware.
Livermore, on a season loan from Tottenham, said: "Of course I want to play. I'm 24 years of age and this is the first run of games I've had for a long time.
"It's a massive thing to look forward to."