Could I still mow?

No point in top dressing whilst it's too soggy. Wait for it to dry out a bit.

I mowed a couple of weeks ago which went OK apart from all the squigged wormcasts.

I don't top dress mine as there's too much weed and moss and it would be too much work. As long as it looks green it's good enough for me. I don't use chemicals to get rid of the baddies as there is too much delicate wildlife growing in the grass (bee orchids, cowslips etc).