No, HP does not determine MPH in a gear, the gear ratio does. Although there are hypothetical ways that HP can seem to make the vehicle go faster in the same gear, it doesn't actually change the speed capacity of that gear as long as the engine RPM is limited.
As noted, the answer is no... or at least not really.
In lower gears, the gear becomes the limiting factor because there's only so many RPM that the engine can perform. In higher gears, the engine does become the limiting factor because the gears allow sufficient feedback and resistance to the engine that it can't keep pushing.
As an example:
Say your in a gear with a 2:1 ratio and you're traveling 60 MPH. At this point, we can say you're getting 'x' amount of resistance from wind, but with the 2:1 ratio, only 1/2'x' is making it to the engine in the form of resistance. If you shifted gears into a 1:1 gear, then the engine would be fighting a full 'x' of resistance.
This is why if you have a very open, high-speed gear (like most cars' final drive gear) your engine doesn't (typically) produce enough power to totally overcome the various resistances that are in play. The clearest example of this is when you're driving up a steep hill in 3rd and you notice how much sluggishness there is. When you down-shift, suddenly you can accelerate much better. This is because in the lower gear more power is being used as torque and therefore less resistance is transmitted to the motor and your engine can now overcome that resistance. You engine didn't get stronger by down-shifting, but the way that the power is being applied did change.
If you're more interested in understanding a lot of the physics that are involved in translating a car's power to its speed, check out this "Advanced" article. I do stress that it's actually "advanced" meaning that if you're not up on physics and the maths, this may be pretty confusing. But if you avoid the equations, there's plenty of text that helps increase the understanding here.
Raising the engine's HP output will (generally) raise the HP curve along the entire second half of the chart. So this does mean that at redline (say 7500 RPM) you’ll have a little more power than previously. This can mean getting a little more “oomph” to the wheels between bounces off the rev limiter, but that will (if at all) barely affect top speed.
When your gear ratio is say 2.5:1, then it stays 2.5:1 regardless of total power output. Your engine is rotating 2.5 times for each wheel rotation though the amount of persistent force is greater. This is why these monster drag cars often have 1 or 2 forward gears. The sheer amount of power lets them blast into motion and keeps persisting against a really wide gear ratio. So:
Is the speed reachable in each gear directly proportionate to the gear ratios
or can this speed be influenced by the amount of power being put out by the engine
Yes. But not directly. Since we're talking about a mechanical system there are variables in the system itself that will affect overall speed. So the speed can be influenced, but only indirectly.
As Solar Mike said in the comments, though, the added power will reduce the time it takes to reach a given RPM. What could be causing the difference is that the added power is pushing you deeper into the red zone - faster - thus achieving a slightly higher actual RPM before shifting. And depending on the gear ratio, even a change of a couple of RPM could create a 3-4 MPH difference.
I'll also add that it varies this entire argument if you're shifting when the tachometer reads redline, or when you actually hit the rev limiter.