Thanks for the feedback! So to start with I warped the cursor to the shape spawn point. However this is not allowed in webGL... Then I found that having the shape move immediately towards the cursor was a bit jarring. So the approach is that when the cursor is close to the shape it will 'pick it up' (no click required).
The above I'm still not fully happy with so I'll throw some time tonight into thinking about a solution as you rightly say it's too easy to accidentally gameover!