I don't state my salary but I heard from tech firms something in the vein of "we can't match a City salary". Sure, I'm ready to take a salary cut for an interesting opportunity.

But do you say that you are willing to take a non-City salary?

Only after a discussion starts (i.e. not in my resume). Salaries are usually not stated for the positions anyway, so this seems like a fair game to me.

Sorry to resurrect after three days... I think it might be worth saying something upfront. I agree that you shouldn't be precise and say how much you're looking for, there's no need to throw away a negotiating point early. But something in the cover letter that hand-wavingly made it clear that you understand that startup salaries are lower might help.

For example, "I'm looking for something that pays me back in work I enjoy, rather simply financially like my current high-boredom, high-remuneration City job".

...OK, that's pretty terrible. But something along those lines, maybe?