And also on one side of my lawn has a tree that has small orange red berries. Will the fallen berries on the lawn prevent grass seeds to germinate? Thanks for your help!

It has been known to help with consistent germination, but it has a lot of disadvantages too, especially when it comes to cool season grasses that don't spread on their own. You'll bring up weed seeds for sure.

I don't like core aeration for seeding purposes, because the seeds tend to wash into the holes and create a clumpy look. Try to create good seed to soil contact by scalping and raking off as much of the dead stuff as possible. A power rake/ dethatcher will make the job easier but it can be done by hand.

May is not a good time to seed grass. In the south May is a month too early. In the north May is 4 months too early. Where do you live? What kind of grass did you seed?

I don't see a need for aeration in your case.

Red berries are very likely to get redder until Christmas and stay on the plant until they ferment next spring. Look for very happy birds in that tree next spring.

Thank you for all your reply.

I mowed the lawn to about 2 inches high, hand rake and dethatch yesterday. I am not going to core aerate then. I am still worry about those fallen berries on the lawn that may kill the grass seeds to germinate. Already pick up as many berries as possible. I think the tree is a Rowan or mountain ash. I am going to lay 1/8 to 1/4 inch of lawn soil before I overseed.

Your soil looks fine to me, no need to add more on top.

You need to cut lower, as low as your mower will allow - aka scalping. The current grass will happily take advantage of the water and starter fertilizer you put down for the seed and grow quickly smothering the seedlings. You will also find the grass will be so tall you will need to mow in about a week, which means trampling all over the seedlings when they've just germinated.