We will know how motivated Liverpool are after they play their next Premiership match because ideal situation is always different from reality. You may be expecting them to be highly motivated to focus on the league without knowing that they have been demoralized by the two consecutive losses that kicked them out of the two competitions which they were even the to win the cup. Such psychological condition can impact them negatively in the Premier League matches and make them lose or draw matches they were supposed to win. Anyways, it is my expectations that it does not get to this point for them so they don't put their chances of winning the Premiership on the line.
After a good start to this season they couldn't cope with the most important games, it can be morally breaking, but in the Premier League they just have to win, because City and Arsenal helped them a lot by dropping points too often. So Liverpool don't need to win everything to become champions, to reduce 12 points in 9 games is possible but for that Liverpool would have to play frankly badly, and I don't believe in it.