Can you put it back quickly before the mom forgets about it and builds another nest?
I would not move it unless I could move it a few feet at a time maintaining about the same environment (amount of shade) each time. Wait a few days and observe that mom has settled onto the nest again between each move.
When cleaning anything including baby diapers, it is better to sweep off or scrape off as much solid and semisolid material as you can first. (1) Less cleaning fluid or detergent needs to be used, (2) Less of a mushy mess which in turn means that bleaches or disinfectants will be more effective.