The planet I have in mind is slightly larger than Mars and is located in the system behind a planet about twice the size of Earth.
The effect being almost like a solar eclipse that happens daily (If that makes sense).
EDIT: I like the idea that they are not double planets but in separate orbits instead. Would this increase the possibility of the planet, which is the smaller of the two, and is further away from the system's star, having longer nights?
You're talking about a double planet (and may want to look at this question).
The two planets have masses of 7M (twice the size of the Earth is about eight times the volume, but I'm allowing for more light elements to have been captured) and 0.15M respectively, so the ratio imbalance is about twice that of the Earth-Moon system; using Asimov's satellite criterion, ours is a double planet system.
If it orbited at the same distance of the Moon, Alpha being 7X as massive as Earth, Beta would complete one revolution in about $\sqrt{7}$ = 2.64 less time, giving 10.6 Earth days. To have it orbit every day, it must be $10.6^\frac{2}{3}$ = 4.827 times nearer, which means about 78720 km between the centers (62600 km between the surfaces), in geosynchronous orbit.
(The Roche critical distance for Alpha is $d = 1.442 R_m$ or about 6000 km above the surface; so, while it would experience some remarkable tides, Beta would not disintegrate into a ring).
Supposing Alpha rotates once every 24 hours, has no axial tilt, and Beta orbits on the plane of the ecliptic, we are in a tidal lock situation - Alpha always shows the same face to Beta and passes directly between Alpha and its sun. Which means that during daytime, Beta eclipses Alpha, and half of Alpha is always dusk (either because it's a night illuminated by a full Beta, or because it's day and Beta causes a solar eclipse), while the other half of Alpha never sees Beta and has a normal 24-hour day cycle.
In this schema, Beta and Alpha rotate around their common barycenter in 1/364th of a 365-day year. So each year Beta "falls behind" by 24 hours, and more importantly, in six months it has fallen behind by 12 hours, and is now on the opposite side of Alpha. From the point of view of Alpha City, Beta moves in the sky by about one degree every day. When it is high in the sky at noon (right side), Alpha City is in totality and has 24 hours of night - so, more like one or two months of night. Then they have several months of penumbra. Finally Beta sets, and for four-five months you needn't worry and enjoy a normal circadian rhythm (while Counter-Alpha City on the other side of Alpha is in the dark); finally Beta is low on the horizon in the morning, prolonging the night with the penumbra of the oncoming eclipse.
Having Beta counter-rotate, or being in a 3:2 or 1:2 resonance instead of a 1:1 lock, or in a different orbit, gives more complex setups with an alternance of "long" and "short" days.
If you want to have long eclipses, you are looking for a modified version of Jupiter Galilean moons.
In their sky Jupiter is pretty large and the Sun pretty small, therefore the solar eclipses shade the entire planet. Since they also orbit around Jupiter, they are granted to get eclipses really often.
In certain locations of the moons one can be on the dusk side of the terminator right when entering the shadow cone: this would make the night last apparently longer than normal.
Logically, no. I can't find direct evidence (yet), but it makes logical sense that this isn't going to be possible.
The question would be how a Mars-sized planet would create a global longer day if the current moon with half it's size can only eclipse about 100 to 160KM (https://en.wikipedia.org/wiki/Solar_eclipse), so the Mars planet would offer a shadow of 200 to 320KM eclipses. Unfortunately your planet is also twice the size of earth so the amount of surface area that can be covered is smaller.
Then we also need to cover the fact that a Mars-sized moon would need a faster rotation around the earth to prevent it from crashing into it, but not too fast because then it would just fly off into space. Even then you still have a rotation of days around the earth, anywhere between 13 to 20 days I would guess (normal Moon cycle of a bit more than 27 days). So only half of those days the Mars planet would be able to create an eclipse and "lengthen the night", even though it wouldn't be able to really create a global eclipse unless the sun it's orbiting is also extremely tiny and far away, which kind of pushes the planets outside of the goldilocks zone.
