I’ve introduced 2 different cats to 7 different dogs, only 2 were puppies right enough.
This is what works with the current cat who is used to dogs but quite a timid cat, as in she’d run and hide rather than fight a dog harassing her.
Just let the cat do what she wants and control the puppy... as in the cat gets free run of wherever she wants to go and the puppy only gets to meet her up close if the cat initiated it.
I leave doors open if they’re rooms the cat usually spends time in, but the puppy would be on a house line if the cat is up and and likely to come in, so I can refocus the puppy before it goes... OMG, friend!! The cat doesn’t spend a huge amount of time in the living room (she prefers the dining room) so that door is mostly closed if we’re in there till the puppy/dog is reliably going to ignore the cat - just because it gives time without having to watch for the cat.
Food, water and litter trays are all places the dog can’t get to.
Loads of treats for focusing on me when the cat is around, I train leave and recalling to their name pretty much immediately.
Once the dog is reliably ignoring the cat doing cat things and I know I can tell it to leave or call it away and it will, I pretty much leave them to it, with the odd, look mate, I know she rubbed against you purring, but she’s not a dog, it didn’t mean she wanted to wrestle, leave her alone.
I find for the first few weeks the cat likes to sit somewhere she can see the dog, but it either won’t notice she’s there or can’t get to her, so you see her peering round doorways a lot or sitting halfway up the stairs watching it. Once she’s happy it’s not going to chase her she gets more interactive.