You need to get the moss out first-use a combined lawn weed,feed and moss killer-then when moss has gone black scarify-then at some point over-sow with grass seed if needed
But you cannot do any of this when everything starts growing-so not yet
You are welcome-there are many knowledgeable people on this forum- someone usually comes up with an answer
Often several people come up with several answers and you have to sort out which one you like best. I shan't be offering an opinion on lawns. I just have grass. and moss and clover and plantain and..............
If your lawn is not soaking wet (like mine!) use a moss killer then scarify. It will probably look muddy and awful to begin with. I mix compost with grass seed and scatter it in early spring or in autumn (too late now). Use an organic fertiliser when the grass is coming up well. You can use a 3 in one as Geoff suggests, but where I live it is quite expensive.
Moss is a sgn of poor drainage or excessive shade, so the usual advice is to try to correct these first, or the problem will return. In my experience most of us have moss at some time or other, if not most of the time, and it's just one of those things to add to the list.
Moss is there for a culmination of things...
Shade usually gets the blame, but it will also be lack of nutrition (we remove food all the time when we mow)
Low, irregular mowing.........Saturated soils........ lack of moss control.
Scarifying removes the plant but not the spores....
When using domestic products, the moss killer part cannot hit all plants, let alone spores, so try to do it AFTER you have scarified...
This way, you open up the sward so that the moss killer can get into the bottom of the sward.
Next time you look, when applying first is that you will only dis-colour the top part of the plant...and there will be many healthy plants ready to carry on!
Hollow tine aerate the soils as much as poss.....dont dress sand unless you can gurantee that the sand particles are the correct size and shape.
Leave holes open for roots to get in and develop.
Scarifying will also allow certain species of grass to thicken back up.
Best way to scarify....start as high as possible...and slowly move it down a tad....
Leave some grass to fill in.....after a week or so, apply a spring feed.....
Best to leave weed control as many of the new young recovery shoots may be affected..