Posted: 20/11/2017 at 23:03
It is a bit late for scarifying a lawn it wont recover and will be bare over winter, and its to late for seeding a lawn it wont germinate, you need a shade mix grass seed for under the tree.
You can aerate with fork or use a hollow tine and top dress with sharp sand / top soil or buy a bag of top dressing , this will help with the moss and compaction.
Moss killer only kills the top inch, so if you have a lot of moss scarifying first and moss killing later would be better. You would need a autumn feed which normally has moss killer or just a moss killer if you were to do it now. Spring and Summer feed are high in nitrogen which stimulate lush green growth, lush green growth and Winter do go together to well.
Personally I would hold back for Spring March - April unless its freezing still. If you cant do it in spring do it now but the lawn won't thank you for it and will look very sad.