I would start by fixing the locations of Richard and Hand (did you mean Hank?), keeping in mind that their positions can be switched, so multiply by 2. Now we are dealing with a linear (non-circular) permutation of the remaining 6 speakers, with the sole restriction: Thomas and Lily cannot sit together. I'd then take total permutations, 6!, and subtract those that place Thomas and Lily together. There are 5 ways to place Thomas and Lily together irrespective of order, and 10 considering order. For each of those 10, there are 4! ways to permute the others. So we have
Answer = 2 * (6! - (10 * 4!)) = 960 ways