It becomes a question of what you count as spending and who you count as homeless so it gets difficult to pin down something everyone will agree on. A couple of posters already linked the Hoover study but obviously not everybody is going to run with Hoover. San Francisco's annual budget includes (these are all approximate) $420m for permanent housing, $60m for immediate shelter, and $120m for homelessness prevention, so we're talking roughly $600m plus a large chunk of the $250m that's budgeted for mental health interventions. So we're talking close to a billion dollars a year out of a $14 billion city budget.

The denominator becomes tricky too, in that SFO has an estimated given-night homeless population of about 8000, about half of whom are rough sleepers (people literally camped out on the street/under bridges). Note that this is a high proportion of rough sleepers compared to most cities where it's about a quarter of the homeless population. That translates into about 32000 people experiencing homelessness at some point in a given year (at least that's the rule of thumb I remember from -- wait for it -- working at a homelessness NGO years ago). So the most naive calculation shakes out to $850m spent on 32000 people, or just north of $27K.

Obviously this has some problems, in that at least some of the homelessness prevention money is hopefully preventing a non-zero number of people from becoming homeless (though in my more cynical moments I wonder). But then again somebody who's only homeless for a month shouldn't need a full year's spending.

TL;DR: it's complicated and there's not a single answer everybody agrees on, but in terms of orders of magnitude it's "tens of thousands of dollars per person per year" in most large US cities, with SF as an outlier on the high end.

(If you really want to get depressed, look at European cities like Amsterdam or Berlin which have homelessness rates approaching double most US cities' and even higher expenditures.)