Before answering, I would like to clarify some things;
- Your score is not how many stones you have captured¹, but how much of the board you control. This means areas where your opponent cannot play without being captured. Details: Scoring
- A stronger player does not generally give a weaker player a handicap of so many points of score (though that is possible), but rather gives them free moves for the difference in their ranks.
- In most events, only a win or loss matters, so players do not try to maximise their score: if they seem clearly behind, they may risk losing a lot more even for a small chance to win; if this fails they are likely to resign. This means that statistics about winning margins in points (which are often not recorded) would not allow reliable conclusions.
There are different ranking systems in different countries and on different Go servers, as described at Rating Systems.
In particular, the European Go Database (EGD) operates an ELO system, and provides statistics here on win probability against rank difference and handicap given.
For example, if I select the period 2015-01-01 – 2022-06-22, it churns for quite a while (perhaps 20 seconds), and then shows several tables of statistics. There one can see, for example, that in games between players 2 ranks apart with 2 stones handicap (i.e. the standard handicap), Black (the weaker player) won 39% of the time (34% for kyu – weaker – players, and 39% for dan – stronger – players). For a 5 rank difference Black won 35% (in all cases) with 5 stones but only 16% with 1 stone (17% for kyu players, 7% for dan players).
I have not studied the statistics in detail, but one thing that stands out is that the standard handicap (1 stone per rank) only gives Black about a 35% chance of winning in most cases, except for a difference of 1 rank, where it is about 45%. Further, to answer your bolded question, it looks as though the weaker player needs 1 or 2 stones more than the rank difference for an even chance, but such a handicap is almost never given, so that is an extrapolation.
(You actually seems to be asking about the points needed to make a game even, but the score difference is often not recorded, and I think that these (EGD) statistics do not allow a conclusion about this. Also the approach I mentioned of playing for a win rather than to minimise your loss would confuse the issue further.)
Secondly, weaker players get better results than stronger against opponents a given number of ranks stronger (also in games without handicap); perhaps this means that weaker players vary more, or that their ranks are less accurately estimated by the system.
¹ This (score = captives) may possibly have been the case in the origins of the game, though the situation seems not really clear to me, and is now definitely not the case anywhere I know of. See Ancient Chinese Rules and Philosophy and Prisoner Scoring for some more details.