I'd like to submit here something "odd" just to check if you consider it worth to be worked out or not. Please be patient for some math, I tried to keep it to the minimum.
Imagine you are a perfect marksman (no aiming errors) and, for the sake of simplicity, you are in the perfect position behind the target (dead six). Both target and your plane fly at the same constant speed along a straight path. The distance between the two planes is D. Finally let us neglect the effect of drag and gravity so that once fired, the bullet moves along a straight line at a constant MV relative speed with respect to the target.
If all this stuff is true, the time the bullet takes to reach the target is TOF=D/MV. If ROF is the rate of fire (bullets per seconds) the time between two successive bullets on the target is Delta_T=1/ROF.
Imagine now that, in the very moment you start firing, the target starts maneuvering to avoid your bullet stream. That is, the target starts to move away (horizontal, vertical, a combination, in a word, "lateral" move) from the impact point. If we assume that the "lateral" acceleration is A m/sec^2, then, according to the basic law of motion of a "body" under the action of a constant acceleration, the distance of the target from the impact point at the time T is given by S=0,5*A*T^2.
The lateral spacing between the aimed impact point and the target, after TOF seconds will be S=0,5*A*(D/MV)^2 and, after each further Delta_T seconds, the spacing from the stream of bullets will be increased by Delta_S=0,5*A*(1/ROF)^2 meters (while firing your "short" burst you don't change the aiming point).
What is, IMHO, worth noting now is the fact that both MV and ROF are "squared". Therefore, for the same maneuvering capability of the target, an higher MV (and or ROF) seems to play a "more than proportional" role in the Hit Probability game. That is, if we compare two guns with two different MV (MV1 vs MV2) and or ROF (ROF1 vs ROF2) , the ratios between the corresponding target " escape" spacings S2 and S1 are proportional to (MV1/MV2)^2 or (ROF1/ ROF2)^2. If " the lower the spacing the higher the Hit Probability" assumption holds (*), then the square of the ratios between different MVs or ROFs should play a role.
(*) Bullet dispersion distribution is not uniform. For the same target "cross-section", as its distance (lateral displacement) from the aiming line increases, the bullet density decreases thus the cumulative probability of hitting the target decreases. Imagine this cumulative probability (HP actually) changes, more or less, as the inverse of the distance from the aiming point. Stating that HP is proportional to 1/S (the inverse of lateral spacing) means HP is proportional to (2/A)*(MV/D)^2: if the target is moving away with a constant lateral acceleration, then, doubling the MV will imply a ¼ factor for the displacement and a factor 4 for HP. Finally Aircraft aren't spherical or ellipsoidal objects. Thus maybe the geometry could play a role in decreasing the HP more "quickly" than assumed, as distance of target from the aiming point increases. If this is true, then increasing the MV could have an higher than "quadratic" effect in increasing HP. By the way, if this last assumption holds, even for targets moving away at a constant speed (typical of deflection shooting against unaware targets), the MV increase would have a more than a proportional impact.
Some remarks on target acceleration/size and shooting distances:
Obviously the amount of acceleration plays a key role: an heavy bomber is quite slow both in terms of speed and acceleration. Shooting at these big and slow things is a job were MV could not be the primary feature to improve if HP were the point. Small&nimble fighters are a totally different affair. This suggests that "actual" quantitative links between HP and MV and/or RoF may depend on target features so that war reports/studies and weapon design guidelines could have been influenced accordingly.
Of course, shooting distance plays a fundamental role. At very close shooting distance from big targets, the above mentioned effects are negligible and for a wide range of MV the HP doesn't change at all: most of the bullets would hit the target anyway. At very high shooting distances, the bullet density is so low that even doubling or tripling the HP means doubling or tripling something that is desperately close to zero. But, if the basics I described have some ground, practical values of HP could reasonably depends more than proportionally on MV (and ROF), for an "intermediate" range of shooting distances.
"Conclusions"
The proposed scenario is definitely basic and relies on several hypotheses. No "claim" to accurately model the actual combat scenario by too simple relations; just the attempt to inject some new elements and see if they have some chance to survive the criticism.
