I fixed the rear carrier. I doubt if it's a permanent fix, because the way it attaches to the the bike near the wheel hub, isn't very good. So I've attached a few spare bolts to the bike for when it works loose again. And I'll give some thought to an alternative attachment method. Maybe use P clamps.
The rear inner tube had rotated in the tire - this means that instead of being parallel to a radius, it was at an increasing angle. That might not sound like a big deal, but I know (from experience) that when that angle gets sharp enough, it shears off the valve and you get a total blowout. So I deflated the tube, wiggled it so that it was in a better place, then reinflated it.
But to reinflate it, I use a battery-driven car air pump, the same one that I use as an emergency battery in the car. And it was almost flat. The reason is simple, it's too old. These lead-acid batteries only last a couple of years, and it's three years old. So I went on to Ebay and bought a new one for £25. I have needed it a couple of times - once when I stupidly left the headlights on while I had lunch in the car, and that flattened the battery. And once when the engine was overheating while I was stuck in a traffic jam, so I was turning the engine off and restarting it, and that flattened the battery.
Did you know that a diesel engine needs more power to start up than a petrol? It means you need a more heavy-duty battery.
I also adjusted the rear brake; it needed tightening up a bit. I oiled the chain, then turned my attention to the batteries.
The new batteries I bought from Hobbyking arrived, and they look good. I also got three more chargers; I no longer use the more complicated balance chargers that mainly charge through the main connector, then balance via the balance ports. Instead, these chargers charge via the balance ports. These are much easier to connect and disconnect, because I don't need to use the main connectors. Also, I've had a few of the big chargers fail (and they don't always fail safe). The chargers I'm using now are only 30 watts whereas the big ones are 80 watts. But I use three small chargers for my battery-triples, instead of the one big charger. So they'll charge in several hours, which is fast enough.
The chargers cost about £6 and I run them from old PC power supplies, which cost me nothing.
... later ...
Yes, P clamps were the answer. I had to do quite a lot of cut-and-try so that they wouldn't foul the chain when in high gear, but I think it's OK now. I feel a lot more confident that when I bump along some of the very rough tracks I navigate, the back carrier, supporting 20 to 25 pounds, isn't going to fail.