Lawn maintenance

I had a new lawn laid last year and have mown it this week for the first time this year. It is slightly ragged in patches and the area that lies in the shade appears to have died.

I have bought lawn seed for shady lawns, and have read  Gardeners' World 12 week plan for a perfect lawn.

Do I now leave the lawn for a week or two prior to scarifying (which is exhausting as the lawn is a fair size) or do I need to take further action during the next week to assist it to its' full potential !!

I have bought chicken pellet fertiliser which I have read needs to be applied in about three to four weeks - If this is the case, do I sow new seed at the same time?