The Reds ended up with just a Community Shield to show for an underwhelming season where they finished fifth in the Premier League, missing out on Champions League football.
After competing until the very end in all four competitions the previous campaign, it was a huge drop off for Liverpool, despite the only summer difference being Sadio Mane departing for Darwin Nunez.
Early exits in all three cup competitions coincided with terrible league form, which was only patched up by a late seven-game winning streak.
That run was triggered by Klopp letting his players know they had to improve.
Asked what the manager's message was to his stars as headed off for the summer break, Robertson told talkSPORT's Jim White: "I think that message started six to eight weeks before the end of the season when we changed stuff and we were always looking to the next season - the message already started.
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"But the pre-season will be a big one. We have to get it right in terms of fitness and everything like that and get ourselves as ready as we can and tactically I'm sure we will be doing a lot of work as well to try and make us a better team and a team than can compete for the trophies we want to compete for."
Robertson concedes Man City were simply too good for everybody as they clinched an historic treble success of Premier League, Champions League and FA Cup.
But the Scot believes Liverpool will come good again and can challenge Pep Guardiola's side.
"City were a class apart from the whole league last season," he added.
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"We weren't anywhere close to that. Obviously the last four, five seasons, we have been the team that has pushed them, we've been the team that has tried to compete with them, but this season they won it all.
"They won the treble and rightly so, they were the best team in Europe by a country mile and deserve all the plaudits they are going to get. But we can't concentrate on other teams, we were below the standards we have set for ourselves and it is up to us to try and get back to that and if we do, I believe we can be successful again."
When questioned about how anyone can knock City off their pedestal, Robertson insisted they would focus on themselves.
"We have to be back to our best and we believe if we are back to our best, we can compete with any team, but if we drop below those standards then we know we are struggling," he said.
"That is what we have done this season - we dropped below the consistent standards we set for ourselves the last five, six years and we have been punished for it. We have to deal with that and try to make sure it doesn't happen again."