Manchester United got a big shock in Champions League that can trigger a domino effect on their next performances. However it won't always happen and sometimes, a team in trouble can strongly bounce back but we won't know how Manchester United will perform in Premier League after disappointed in Champions League.
In general you are right that a team may make come back in local league after bad result in bigger league or vice versa but Manchester United seems to be a different case. They have not shown their best performance yet in both competitions. After they were eliminated from UCL, they have to face Liverpool in the next match. Unfortunately they have to play without their captain due to yellow card accumulation as well as some players who may not play due to injuries such as Luke Shaw and Maguire. Some other players may not be able to play as well such as Rashford and Martial unless these players can get quick recovery. It will be hard match for Man United to set the best possible lineups and strategy against Liverpool.
Right now Ten Hag is under increasing pressure after overseeing three defeats in Manchester United's last four matches, including Tuesday's loss at home to Bayern Munich that saw them exit Europe. Manchester United isn’t not a team that can pull up surprises at Anfield. They have injuries and and their captain is suspended. Even at that Liverpool don’t easily lose when they play at home.
I think Manchester United have changed their manager like 3 times since they last scored a goal at Anfield: five years ago this weekend when Lingard scored in a 3-1 defeat at Liverpool that spelled the end for Jose Mourinho. That's the only time Manchester United have scored at Anfield in their last eight visits in all competitions. And Liverpool have scored the last 15 goals between these teams at Anfield. So I am not expecting anything from them in this game.