@Viopia I think, for the purposes of this question, it would be sufficient to know that the recessional velocities and local 'real' (or peculiar) velocities can differ in important ways, but can also be compared in some meaningful ways too.
In particular, it's worth reassuring yourself that if you were to put a really long and rigid ruler between yourself and some galaxy receding due to the expansion of space, you would definitely see that galaxy speed past the ruler to ever farther distances. You could also add the local radial velocity of the galaxy (or even a photon) to the recession velocity, just like you would add regular velocity vectors, to obtain a nett velocity that would tell you whether the extra local motion makes the galaxy approach, recede or hover at the same proper distance. I.e. there's an unambiguous physicality here, that you can depend on to visualise what's going on.
At the same time, it's good to keep in mind that there's enough of a difference between the two, that naively extrapolating any further intuitions about velocities should be avoided when thinking about recession.
For example, to bring this into the vicinity of the setup in your question, you can have a rocket moving at such a peculiar (i.e. local) velocity, that the nett approach velocity after adding recession velocity is zero. I.e. the rocket can 'hover' at a constant proper distance from the observer. Just like in your setup - though not for that particular supernova, as it was too far and receding too fast.
For a photon, while locally it always moves at c, it can have any approach velocity whatsoever - including 0 and negative, i.e. being 'stopped' or carried away by expansion despite being sent towards us.
But - here's an example of an important difference - it turns out the redshifts from peculiar velocities and recessional velocities do not add up in the same way. It's not valid to simply add up the two velocities if we want to know the resultant redshift.
A signal, made by the rocket to have the same form at emission as the signal emitted in its rest frame by the supernova it attempts to mimic, and emitted by a rocket with zero approach velocity at the same place and time as the supernova, would not have nett zero redshift, nor would it be redshifted - but would be blueshifted instead.
The details of how this works can be seen
here, section III. The figures 5 and 6 can be helpful. (the line marked =(0,0) describe the unaccelerated universe we've assumed here, while =(0.3,0.7) is likely our universe).
So, you would have two photons - one from the supernova, redshifted; another from the rocket, blueshifted. They are therefore distinguishable.
Both photons would arrive on Earth roughly at the same time, as they have the same distance to cover, contrary to your intuition expressed above.
This can be understood by imagining what happens locally, in the immediate vicinity of the supernova. If we arranged for the rocket to emit its photon as it's passing the supernova, we would end up with the two photons traveling together. Since they travel together, in whichever way the expansion affects the intervening space they have to cover, it will do so in the same way for both photons. And since they travel together, and both travel at c, both arrive at the same time, having covered the same distance.
As is mentioned in the article linked earlier, this situation can be observed in nature, with relativistic jets from active galactic nuclei acting out the role of the rocket moving towards Earth, while the faraway galaxies hosting the AGNs are playing the supernova in this scenario.