The user is to draw a polygon in the map frame. The first point is drawn fine. For the second point it may take 8 to 10 clicks before the point appears on the screen. This doesn't happen in debug mode in VS 2010.
Are you trying to do both or are you saying that the Draw in MeasureAction takes multiple clicks before second point is drawn? This behavior does not seem reproducible with the two samples above. So you might need to check your application logic, maybe something in there disables the draw. Also, the published website should not be any different as what you see in Debug mode if they both point to the same Silverlight application.