Does Razz Berry work for the next throw or catch?
I don't have evidence for this, per se, but I believe that it applies only to your next successful throw -- i.e., your next throw that actually hits the Pokemon.
I suspect this because you're unable to use a second Razz Berry immediately after using the first, or if you miss the Pokemon a bunch, but as soon as you make contact and it breaks free, you can use another Razz Berry.
To clarify:
On a failed throw, you throw a Pokeball but completely miss the Pokemon. This does not use up a Razz Berry.
On a successful throw, you throw a Pokeball and hit the Pokemon. It wiggles around a bit, and you either catch it or it breaks free. This uses up the Razz Berry, whether you catch the Pokemon or not.
If you want to boost your chances, then every time you hit the Pokemon with a throw, you'll need to use another Razz Berry.