Moss and lichen in lawns is caused by poor drainage and poor light

You can improve the drainage by spiking the lawn with a fork as deep as you can and waggle the fork about to widen the holes and sprinkle horticultural sand down the holes to keep the holes open. You can get a tool which looks like a fork but the forks have hollow sections so the soil is removed.

You can rake out the moss etc and other unwanted stuff and use a moss killing weed killer and reseed the bare areas.

Unfortunately I think you are too late to this this year as the grass has almost stopped growing.

Do all I've suggested once the grass is growing strongly in spring .

If the shade is caused by trees/shrubs you might want to consider cutting these back to allow more light to the grass. You can also buy grass seed which has been specially selected to grow in shady areas.

A couple of other things. Mowing lawns too close can also lead to problems as the grass is not long enough to photosynthesise enough. In spring and autumn raise the height of your lawn mower a bit.

Finally, it is possible your lawn is short on nutrients so give it a granular feed( spring formulation) once it starts growing strongly in spring and an autumn formulation feed in late September to Mid October.