I've used Evergreen No Rake lawn moss remover, which has bacteria to eat the moss, and organic feed. It worked well, and it also reduced and weakened the thatch layer, which had got quite thick and tough. It's pellets that you spread over the lawn. Our dog found them quite appetising for the first week or so, until the smell had gone, so we had to keep him off the lawn.
Verve do a pack of organic soluble lawn feed capsules; you dissolve them in the watering can, which is a bit slower and messier than you might think, basically your dissolving a compressed powder into water, so you get some lumps, but they greened the lawn up nicely, so I've got some more of those in stock. I found it easiest to dissolve them in a jug first, before adding to the watering can.
G&R