Given that $ (4,7) $ is on the graph of $ y=3f\left(2x\right)+1 $, we can substitute $ x=4 $ and $ y=7 $ in that equation to obtain $$7 = 3f\left(2\cdot4\right)+1.$$We can rewrite this information as $$2 = f(8),$$which tells us that $ (8,2) $ must be on the graph of $ y=f(x) $. The sum of coordinates of this point is $ \boxed{10} $.