I just gently dab dry with something like a cloth or even the soft bedding. If it is dried, I don't worry about it. I believe the less you mess with incubating eggs and the broody the better.
Timothy hay that you get at the feed store in small packets makes a great clumping poo gatherer. I will often use that on top of the nest for that purpose.
PD-Riverman is correct to be sure that food and water requires the hen to get up and take at least a step or two away from the nest. That often helps with preventing nest soiling (but not always, some ladies are less than tidy no matter what).
I've had plenty of chicks hatch healthy and fine from soiled conditions. While not preferred, the inside of a chicken coop (and the backside of a hen for that matter) is hardly sterile.
Nature has supplied healthy chicks with the bloom of the egg and then fully closed navels at hatch that protect them. It is when the navels don't close fully in hatch when omphalitis (infected navels) can set in. Industry attempts to eradicate that with sterile hatching conditions of which we can never hope to maintain.
So generally, I don't worry about anything that can't be cleaned with a dry gentle dab.
Thank you! I did scrape some off and just left the rest.
So it seems that what I did, thinking it would help my broody, may have actually caused the problem. I put food and water within reach from the nest... I went down yesterday and moved them to the opposite side of the broody hut. Hope that helps...and that she dies actually get up and go get some!