correctable error ?
 

H1 is fouled. The table states that is team foul 7 on V. H1 shoots 1-1 and makes both. V1 inbounds the ball and is fouled. That is the 8th team foul on H so V1 will shoot 1-1. While reporting the foul the official is informed that V really only had 6 fouls and H1 should not have gotten a 1-1.

2.10.1h(e) is very similar and states to cancel the made free throws by H and resume play with V having the ball. In this case I would think that to mean resume with the V free throw.

Is that correct? Does H lose their free throws and not get possession of the ball?

Yes. One of the quirks of the rules -- and, perhaps, intentional so both sides have an incentive to "get it right" before the error happens. What looks like an advantage ("haha -- we get FTs when we don't deserve them") can turn into a disadvantage when it's discovered ("we lose the FTs and the ball!?")

I have never bought this line of thinking and never will. It is just a poor resolution piled on top of bad officiating and table communication.