Answer: Unless you remove the weeds by hand or kill them chemically before reseeding I'm afraid the weeds will overtake the lawn before the new grass has a change to grow strong. I'd remove the weeds, power rake and remove the debris, then reseed the lawn. Grass seeds germinate in 7-10 days when kept watered and in 3-4 weeks the lawn should be thick enough to crowd out any new weeds that try to pop up.
It's difficult to get grass to grow in shade so be sure to look for a shade tolerant grass seed mixture.
Best wishes with your lawn.
Q&A Library Searching Tips