The primary use of Pushes is to remedy Problems - to keep them from piling up an making your life hard during the game, and to keep them from resulting in a downbeat ending (p.26):
To counter a Problem, you must do something that would credibly get it out of your way. This may require a successful Test or the expenditure of a Push or ...
It turns out I misunderstood the nature of the core Cthulhu Confidential rulebook. It contains only one scenario for each protagonist, instead of one campaign per protagonist, like I thought. So there is no campaign to commit to in the first place.
On the other hand, having played the Dex Raymond scenario High Voltage Kill,