Liverpool lack leaders on the pitch who can change the way the team plays. In training the team had clearly set up to play from the back to draw out Burnley attackers which would in turn leave space in the midfield to work in. However this didn't work at all because Henderson was awful. Trying to play out of the back with the two CBs miles apart left acres of space in the middle of the field for the 2 Burnley forwards to poach the ball in - which is why they scored.
Once Burnley were 2 up (and arguably 1 up), there was no space in the middle of the field for Coutinho et al to work with. At this point Liverpool should have had the smarts to change strategy a use the wingbacks to get in behind and lob crosses in and punt balls in to the box from the half way line. The problem with doing this is that you lose possession against a good team and get caught on the break, but Burnley were so deep they would have cleared the ball and given it straight back to Pool. And actually giving Burnley the ball would have given Liverpool the opportunity to regain possession when Burnley didn't have 8 men behind the ball.
At half time Klopp should have taken out Lallana (who is virtually useless once the opposition sits back) and played Origi for some height in the box and put on Moreno so that we had pace on the left (which might have induced some fouls which were lacking from Burnley) and able to whip it in off his left foot.
Henderson was appalling and I can't see how he can start ahead of Can (don't think the 2nd goal would have been conceded had he been on the pitch) or Lucas.
Hopefully this game is an eye opener for Klopp that Liverpool need to learn how to play against a team that sits back if you want to make CL. It is all fun and games gegenpressing in Cup matches where your opponents are forced to try to win, but in the league when you need to grind out wins against Watford, WBA, Sunderland, Burnley, Hull, Leicester you need to have a system to beat teams that sit back. Liverpool need to learn how to get balls booted in to the box. Even if you don't score headers, there is a chance they'll be a hand ball, or a dodgy penalty given, or penalty for shirt pulling. To emphasise just how stupid Pool were against Burnley, when they had corners, they played them short! And rather than running into the box and maybe drawing a foul to earn a penalty, they played the ball out to the centre of the pitch.