I don't think that City are so fed up with EPL titles that they are ready to put the Champions League higher in their priorities. Maybe if they had a guaranteed choice between these two titles, they would choose the Champions League, but there is no such choice, and even reaching the final, you can lose everything. In terms of reliability, the EPL looks much better, especially when your team is in the lead, albeit by a small margin.
City and Liverpool both have busy schedules, they have to play in 3 different competitions, UCL, EPL and FA Cup. Perhaps, the trophy that is not a priority is the FA Cup. the rest they will try to be able to bring the remaining two trophies to the clubs. I'm not sure Liverpool only want to focus on one trophy, neither does City. They will focus on 1 title if they have been eliminated from the FA and UCL. if not, it looks like UCL and EPL are targets for them.
I think the first priority is the EPL. As for any major team from the top 4 championships. Titles in championships starting from 5th are evaluated differently and I am sure that any champion of Portugal or France would give 2-3 titles for the Champions League.
The second priority is the Champions League. This cup is valued less than the national championship, but only because this result is somewhat random.
The third priority is all other national cups and stuff, such as the club world championship. This is nothing more than a nice bonus to the main achievements, but if the main titles are lost, then such titles do not change anything.
Now it has become a priority the Premier League for Manchester City as when they were leading by like 6-9 points from Liverpool I am pretty sure their first objective was the Champions League but know that there is only one point difference I think both Liverpool and City should focus much more on the Premier League.
I still think that the fans of Liverpool would not be disappointed though if they manage to win the Champions League this year even if this would cost them the Premier League title,lots of beautiful memories from the many finals they have already won and adding one to such achievements I am sure they will like it,of course it would be much better to win both trophies but this is somewhat more difficult.