There's a moss forum here somewhere with a faq (see link).
I think moss likes moisture and low light, and most moss do like acidic soil. Maybe you should test the pH of your bed?

Here is a link that might be useful: Moss FAQs: Moss on Soil and Other Surfaces

Moss can be a sign of poor drainage, soil compaction, pH problems, insufficient sunlight, and other issues. I suggest that you find out where you can have a soil test done to check the pH, and then evaluate this site with these other issues in mind.

As rhizo says moss likes wet, acidity and compaction. Based on UK conditions I personally would not bother with a soil test. very few gardeners here have naturally particularly acid soil. And in A raised bed it is even less likely. I would just thoroughly dig the whole bed over (without standing on it and not when it is very wet) and incorporate andything you can get in the way of compost, leaves, etc. My garden has moss in places this year. We have had two sopping wet summers and low light levels. I read in the paper last week that this year there was more sunlight in March and December than we had in August. If this strategy doesn't improve things look at the site of your bed. Is it in shade? What is the underlying soil? did you loosen that when you put down the raised bed?

Moss is a first stage plant that Ma Nature uses most often to increase the level of organic matter in the soil so other, higher order plants can grow. I have moss growing on concrete blocks, concrete walkways, crushed dolomitic limestone, very dry sand in full sun as well as wet, shady soils and the only commonality is a lack of organic matter.
The only way to know what the soil pH is would be with a soil test, and that same soil test could give you some information about other nutrients in that soil. Along with that good, reliable soil test (just where in the UK to get that done I am not aware) would be these simple soil tests,
1) Structure. From that soil sample put enough of the rest to make a 4 inch level in a clear 1 quart jar, with a tight fitting lid. Fill that jar with water and replace the lid, tightly. Shake the jar vigorously and then let it stand for 24 hours. Your soil will settle out according to soil particle size and weight. A good loam will have about 1-3/4 inch (about 45%) of sand on the bottom. about 1 inch (about 25%) of silt next, about 1 inch (25%) of clay above that, and about 1/4 inch (about 5%) of organic matter on the top.

2) Drainage. Dig a hole 1 foot square and 1 foot deep and fill that with water. After that water drains away refill the hole with more water and time how long it takes that to drain away. Anything less than 2 hours and your soil drains too quickly and needs more organic matter to slow that drainage down. Anything over 6 hours and the soil drains too slowly and needs lots of organic matter to speed it up.

3) Tilth. Take a handful of your slightly damp soil and squeeze it tightly. When the pressure is released the soil should hold together in that clump, but when poked with a finger that clump should fall apart.

4) Smell. What does your soil smell like? A pleasant, rich earthy odor? Putrid, offensive, repugnant odor? The more organic matter in your soil the more active the soil bacteria will be and the nicer you soil will smell.

5) Life. How many earthworms per shovel full were there? 5 or more indicates a pretty healthy soil. Fewer than 5, according to the Natural Resources Conservation Service, indicates a soil that is not healthy.
and these will help you determine what needs to be done to make that soil into a good, healthy soil.

Many thanks for the advice. I've removed the polytunnel, dug it over, and spread compost over the top. Fingers crossed!

Charles

Charles - that's a good start. You didn't tell us about the polytunnel in the first post. That would be adding to your problems of poor air circulation which can also encourage moss. We are very lucky here in the UK in that few gardens are situated on really awful soil unless you are in a new build where the top soil has been removed/turned under. In the States people seem to have to contend with a lot more difficult soils and tests are apparently more used. If you spend much time on these forums you will see a lot more preoccupation with 'amendments' (a term I had never heard before coming here) and trying to alter the native soil than in UK sources. We are really lucky in that most UK garden soils start reasonable and are relatively easy to improve with organic matter and a bit of attention.