What's the significance of HUP if Ψ collapses?

[Phys12]

If I understand it correctly, a particle doesn't have a definite momentum and a definite position, but is in a superposition of multiple positions and momenta. And when we measure either of the two quantities, say, position, the wavefunction collapses to tell us where the particle is. Now when we measure the particle again, it will always be in that spot since we have changed its state. If that's true, then what exactly is the use of the uncertainty principle? If the measurement changes the state, wouldn't measurement of the other quantity be irrelevant? Or does the uncertainty principle tell us that the spread of the various positions of a particle is related to the spread of the various momenta that a particle can have?

 

[Nugatory]

Or does the uncertainty principle tell us that the spread of the various positions of a particle is related to the spread of the various momenta that a particle can have?

Yes.

We can prepare the particle in any state that we please, and having done so we can measure either the position or the momentum. Suppose we prepare the particle in such a way that the result of a position measurement will be somewhere between $x$ and $x+\Delta{x}$; for example we might put the particle in a box of width $\Delta{x}$. In principle we can make $\Delta{x}$ arbitrarily small, so we can measure the position out to as many digits as our lab equipment is capable of.

But suppose that after having prepared the particle in such a state, we choose to measure the momentum instead? In principle we can measure the momentum to as many decimal places as our lab equipment is capable of, and we can always buy better lab equipment if we want even more accuracy.
If we do this experiment over and over again, we will find that our super-accurate momentum measurements are spread out between $p$ and $p+\Delta{p}$. The uncertainty principle is a relationship between $\Delta{p}$ and $\Delta{x}$; we can prepare a particle in which one of them is arbitrarily small but then the other will be commensurately larger.