After a little checking around, I discovered that it wasn't simply a washer issue from a screw-thread connection, but the metal reticulated hose itself that had sprung a leak. No biggie, we'll just get another. In the meantime, I noticed that it tended to leak more when it was at an angle, so I detached the hose and turned it around so the leaky end was now attached to the wall instead of to the handle (so there was less wiggling and thus less leaking).
The new hose came in today. Not just a new hose, but a bunch of little fiddly bits that could replace perfectly fine parts with what we currently have. I break out the new hose and find that it's just plain not going to work -- the previous hose ended with two female ends, each the size of the sort of thread that screws onto a beer keg. The new hose ends with two male ends, one bigger and one smaller, neither as big as the female ends on the old one.
Initially, I called foul and said it wouldn't work, but I was cajoled into trying it out with all the (un-needed) new parts. Now, what I haven't mentioned is that there's this brass ring that can slide down over the handle so you can leave it spraying without having to actually hold it. The new parts include a new handle, which is now too wide to accommodate the ring. I'm told, "You can get a new one of those rings anywhere!" to which I reply, "Really, so I can hit the 7-11 on my way out of work and get one?". *ahem*
I'm convinced to try it with the new parts for a bit and to deal with the fact that the ring no longer fits. First thing I notice is that the water pressure is probably half of what it previously was (probably due to a narrower pipe in the new parts). Not long after I notice that there's water dripping from the to-wall connection. No, it's not leaking from the hose (like the old hose was), but instead it's leaking from a screw-thread connection that they didn't bother including a little washer for.
Taking a deep breath, I carefully undid all the new parts and set it up the previous "broken" way, since the "fix" not only still leaks, but adds two new problems.
And this is why I hate foodservice machinery repair-folks -- it's virtually always like this when something needs a replacement part.