Can I edit my route in ForeFlight on the web by clicking and dragging the route?

Yes. Simply click on the route and drag it to a location that you want in your route.  When you release your mouse button, a pop up will appears with options to choose to add a lat/long value, nearby waypoint, NavAid, or airport ID.

Image 1.  Pop up presented after dragging and dropping the route.

Also, if you sync your ForeFlight on the web route to ForeFlight Mobile on your iPad or iPhone, you'll be able to tap-and-drag it within ForeFlight Mobile. Any changes you make within ForeFlight Mobile on your iPad or iPhone will then be sync'd back to ForeFlight on the web.