In your database for hoppip, you have it not learning any HMs. That may apply to the G4 games but I was able to teach it Flash in FireRed. I just realized that Flash is one of those HM\TMs that flip flop depending on the gen. You can ignore this if you choose.
On the pokedex pages it's only generation 4 and 5 moves. Flash is listed under TMs since it was a TM not an HM in generation 4.
When I do the generation 1-3 movesets it will be added under HMs.
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