It’s amazing how well a lawn will come back with some tlc.
Sandy soil will be lacking nutrition hence moss outcompeting them grass.
First thing to do is kill the moss. As mentioned, powdered iron sulphate is really cheap. A 25kg sack kept in a dry place will last you a lifetime. Apply as directed. Just take your time getting the dilution right then write it on the watering can in permanent marker. Sorted forever. You can do this now as you want cool, damp weather.
Leave it two weeks then hire a scarifier. Bribe friends to help rake up and get stuck in. Aim for at least 4 passes in different directions and don’t go deep into the soil.
Once it’s all piled up somewhere it’ll break down and reduce to pretty well nothing over the year.
Apply another dose of moss killer. This will properly get to all the little bits that remain.
If you moss kill once or twice a winter then you’ll never have to scarify again.
Next, learn how to feed your lawn. A rotary spreader is nice and buy fertiliser in 25kg sacks, it’s way more economical. Little and often is better than a load all at once.
Just buy the one called ‘spring and summer’ for now.
Once a year you can use a weed and feed granule. Around May is a good time. Never apply in drought conditions.
That should do it. Each individual blade of grass can spread around 6 inches over the summer. It’s rare to need to re seed.
I’m a professional gardener btw. Been doing this for decades.