(I have already written an answer but I felt it only complicates things; Hopefully, this new attempt will be better)
(In this answer's terminology it is the Sun that moves around the Earth.)
There are two factors that determined the time the Sun spends above the horizon:
- how much it moves Eastward in RA - the longer it goes the longer the day.
- How much it moves away or towards the celestial equator (i.e., toward declination 0): if it moves away it makes the day longer. if it moves towards the day is shorter. (Let's assume we deal only with Summer days where the declination is Negative in Southern Hemisphere and Positive if the North)
When the Sun moves faster (when near the perihelion) then it means both factors are amplified; but the amplified second-factor effect might have a positive or negative impact on the length of the day.
If that was not enough, One should not that those two factors are not independent: bigger movement in declination means smaller movement in RA and vice versa. this has nothing to do with the "speed amplifier" mentioned above.
There is yet another big catch: the influence of each factor varies as a function of the latitude we are in. in the Equator only factor (1) counts and at the poles only factor (2) counts. (this is because at the Equator the declination circles are perpendicular to the horizon, hence movement in declination won't change the altitude or the day length).
Let's try to look at a specific example our day will be 26 Jan in Southern latitude. when the Sun is quite fast (the perihelion 4 Jan). We should consider our two factors. First, we should note that the internal division (as we said there are dependent) makes the RA component stronger on this day than average. So far for Factor (1) the RA has very strong movement to make the day longer! (both the internal division and the speed - it could be seen in this graph, thanks to @PM2Ring - both dash lines are low)
But as for factor (2) it has a negative impact on the day length since after 21 Dec the declination goes closer to the equator. On one hand, this factor is mitigated, as we said, on account of the internal division, but on the other hand, it is amplified because the Sun is fast.
So we have conflicting influences of our two factors. On small Southern latitudes, clearly, the winner is the first factor resulting in a longer Summer day, but as we go closer to the South pole it becomes less obvious.
Let's take a look at Peter I Island (latitude 68 South) rise and set time in January. On 26 Jan the day length becomes shorter by 15 minutes or ~900 seconds. we are faster than the equivalent day of Norther Summer by about 2% that's about 18 seconds. The RA gain must be very small on this day (on the equator it makes the day no longer than 8 seconds; so in Peter I Island, this effect is probably around 1 second.). So overall on this day in Peter Island, the faster movement of the Sun actually makes the day a little shorter by several seconds.
The point being made in this long post is that this is not all about the speed of the Sun and there are other factors the determine the length of the day.