Novice Lawn problems

Hoping for some lawn improvement advice....

I have been trying to improve my lawn over the last few months. It seems to have poor drainage and a significant moss problem.

i have scarified it twice over the last few weeks, and aerated with a fork. I've put lawn sand over the area also.

Now in the scarified 'bare' areas, weeds are multiplying! What should I do next? Do I keep scarifying or is this detrimental? should I put a weed and feed down?

help!