It would be best to rake it in, and make sure it is kept moist.

However, I don't have a rake, as my garden is so small, it would be a waste of money. I usually sprinkle mine on top and it does fine, but the trouble is, you are in real danger of birds having a nice feast.

If you are willing to splash out, and want a fresh new lawn, you could always invest in some turf.

Depends totally on what you want. Patch up or new fresh lawn.

And welcome to GoY

You can't do anything while its a sea of mud - you need the soil to be what's called friable - which means a bit like a crumble mixture - dryish, able to be raked up to a fine tilth of a couple of inches deep. Once you've achieved that, either by raking or digging lightly with a fork, level it off and apply the seed - use plenty so the birds can have some. Keep the dog off for at least 2 or 3 months - once it starts growing, its really easy to uproot it with foot traffic, never mind dog's paws. First cut should be with hand shears if its a big patch, but if you have a hover mower, you can do it with that when its long enough. Best done March if possible, but April will do, it'll germinate quicker then anyway - keep the seeded areas moist if the weather's very sunny and dry.

♥ Thank you both for your advice and comments ♥