There is a similar question here: Can a planet have a day that's always longer than night?1
Some of the answers might be useful to you if you can find a way to reverse the situation.
Added Jan. 07 2019. I add this quote from my answer to a similar question:
There is a simple geometric reason why normal stars illuminate half of a planet's surface at any one time.
The planet Earth has a diameter of 12,742 kilometers, the Sun has a diameter of 1,391,400 kilometers, 109.19 times as great. So if the Sun and the Earth were touching, the Sun would illuminate a lot more than half of the Earth's surface at any one time. The farther away the Sun was from Earth, the smaller the proportion of Earth's surface it would illuminate at any moment, But even at infinite distance the Sun would still illuminate at least half of the Earth's surface at any one moment.
So any normal Earth like habitable planet orbiting any normal Sun-like star at any normal distance necessary to be in that star's habitable zone is going to have very slightly more than 50 percent of its surface illuminated by that star at any one moment.
Could this planet exist?2
So The problem here is that given the probable and many would say possible differences between the size of any possible habitable planet and its star, one half of the planet would always be illuminated by its star. Since the star of any reasonable habitable planet should have tens or hundreds of times the diameter of the planet, the star's radiation would actually illuminate slightly more than 50 percent of the planet's surface at any one time.
Any habitable planet would also have a rather dense breathable atmosphere and atmospheric refraction would cause the sky to remain bright even when the star was a few degrees below the horizon. Thus dawn and twilight would make day last slightly longer than night.
Of course if the planet, like Earth, has a significant axial tilt, there would be some regions on the planet where, during 2 brief periods each year, there are 22 hours of night and 10 hours of day. There actually are places on Earth near the poles which have 24 hours of daylight each day during the middle of summer and 24 hours of darkness each night during the middle of winter, so there are certainly places on Earth where daylight sometimes lasts for 0.3125 of a 24 hour cycle, or 7.5 hours per day - but only for 2 brief periods during each year.
Having the planet's daylight last for merely 0.3125 of its rotation period seems almost impossible to me, since the star that illuminates the planet should be far away from the planet and have many times the diameter of the planet.
So, in desperation, I suggest that the planet could somehow be very close to a very faint and very small star that actually has a significantly smaller diameter than the planet. Thus at any one moment only less than half of the planet might be illuminated by the star.
A habitable planet could be significantly larger than Earth, though there are limits to how large a planet could be and still be habitable.
But even the smallest red dwarf star should have a diameter several times that of Earth, and thus unlikely to be significantly smaller than that of any plausible habitable planet.
White dwarf stars typically have diameters between 0.8 percent and 2 percent that of the Sun, compared to Earth's diameter of 0.9 percent of the Sun. So a very small white dwarf could have a diameter 0.8 percent that of the Sun and could be orbited by a planet with a diameter equal to that of Earth, 0.9 percent of the Sun's diameter, or slightly larger, perhaps 1.0 percent or the Sun's diameter, or 1.5 percent of the Sun's diameter, etc.
It has been proposed that white dwarfs with surface temperatures of less than 10,000 kelvins could harbor a habitable zone at a distance of c. 0.005 to 0.02 AU that would last upwards of 3 billion years. This is so close that any habitable planets would be tidally locked.
If you want your habitable planet to have days you don't want it to be tidally locked with one side always facing the star and illuminated and one side always facing away from the star and in eternal darkness.
And a habitable zone around a white dwarf star of about 0.005 to 0.02 AU would be about 747,989 to 2,991,957 kilometers, or 464,799 to 1,859,116 miles. Even if the habitable planet has a diameter of 2 percent of the Sun's diameter, or 27,820 kilometers, a distance of as little as 747,989 to 2,991,957 kilometers would be 26.88 to 107.54 times the diameter of the habitable planet. At that distance almost a full half of the planet should be illuminated by the star at any one moment.
And there is the problem of how a planet would be orbiting the white dwarf star in the star's habitable zone, which seems rather dubious. As far s I know, the only way a habitable planet could exist in the habitable zone of a white dwarf star would be if a super advanced civilization put it there.
So a naturally habitable planet with a naturally shining star close enough to illuminate only 0.3125 of the planet's surface at any one time seems extremely improbable.
Therefore the situation should be an artificial one. There could be a rogue planet in the icy cold depths of interstellar space which has been given an artificial "star" to illuminate it by a super advanced civilization.
This artificial "star" or "sun" would be a vast artificial satellite orbiting the planet and containing countless vast fusion power generators that power countless lamps on the surface of the satellite pointed toward the planet and illuminating and heating the planet.
Any planet would have a natural rotation period. So the orbit of the sun satellite would have to be calculated so that orbital period and the planet's rotation period combine to put the sun satellite directly above a specific spot on the surface every 32 Earth hours.
For daylight to last 10 out of 32 Earth hours, or 0.3125 of the total 32 hours, the star satellite would have to illuminate 0.3125 of the radius of the planet at anyone time. That would be 112.5 degrees instead of 180 degrees.
I guess simple geometry would show what height the satellite would have to orbit at in order to illuminate 112.5 degrees of the surface at any one time.
And simple orbital calculations will show at what height the satellite would have to orbit in order to compensate for the natural rotation period of the rogue planet to produce a 32 Earth hour daily cycle.
Then calculations will be needed to modify factors such as the mass, radius, and density of the rogue planet and its natural rotation period so that it is possible for the height necessary to illuminate 112.5 degrees o the planet's circumference at any one time to be identical with the height necessary to have a 32 earth hour daily cycle.
If the sun satellite illuminates 112.5 degrees of the planetary circumference at any one time, the illuminated region will reach 56.25 degrees on either side of the sub satellite point. If the sun satellite has an equatorial orbit the regions more than 56.25 degrees north and south of the equator will never be illuminated or heated by the sun satellite and will have eternal night.
Regions between the equator and 56.25 degrees north or south with have lengths of daylight ranging from 10 Earth hours per 32 Earth hour day at the equator to zero hours of daylight at 56.25 degrees north or south.
If the sun satellite is moved into a somewhat higher orbit around the planet, there can be a orbit where the planet as a whole averages 10 Earth hours of daylight per 32 Earth hour period, with places closer to the equator having longer amounts of day and places farther from the equator having shorter periods of daylight.
There might be a possibility of having a natural star with a natural daylight period of 0.3125 of a rotation period.
There might be a layer in the planetary atmosphere, and/or a zone of particles of some kind in orbit around the planet, that reflects the star's light when it hits at an angle of less than 33.75 degrees above horizontal. Thus sunlight will never reach the ground of the planet except when the planet's star or sun is more than 33.75 degrees above horizontal, and the illuminated period should last only 0.3125 of the total 32 Earth hour rotation period.
Another possibility is if the "larger near by planet" mentioned in the original question is a gas giant planet, and the planet in the story is actually a giant Earth-sized habitable moon of the gas giant planet.
The planet sized moon would be tidally locked so that one side of the planet-sized moon always faced the gas giant planet and one side always faced away from the gas giant planet.
If the planet-sized moon orbited the gas giant planet with a period of 32 Earth hours the side that faced away from the gas giant planet would face the star and be illuminated and have daylight for half that period, or 16 hours, and face way from the star and in be in dark night for 16 hours.
If the planet-sized moon orbited the gas giant planet with a period of 32 Earth hours the side that faced toward the gas giant planet would also face the star and be illuminated and have daylight for half that period, or 16 hours, and face way from the star and be in dark night for 16 hours. Except...
Except that when the planet-sized moon was on the side of the gas giant planet away from their star, and thus the side of the planet-sized moon that always faced the gas giant planet was also facing the star and illuminated, the planet-sized moon could pass into the shadow of the gas giant planet and have an eclipse that might last for hours.
Such an eclipse by the gas giant planet would have to last for six hours to turn sixteen Earth hours of daylight into only 10 Earth hours of daylight, five hours before the eclipse and five hours after the eclipse.
The giant planets in our solar system have diameters of about 49,244 kilometers (Neptune), 50,724 kilometers (Uranus), 116,646 kilometers (Saturn), and 139,822 kilometers (Jupiter), and it is possible for giant planets to have somewhat greater diameters than Jupiter.
Red dwarf stars have diameters down to 0.8 percent of the Sun's diameter, and thus down to about 11,131 kilometers. Thus a giant planet can have a diameter several times that of its star if its star is a very small red dwarf, which means that the shadow cast by the giant planet will expand with distance instead of contract. That is certainly good news for the length of the eclipse on the planet-sized moon.
So perhaps it might be possible for a planet-sized moon orbiting a giant planet to have eclipses long enough for places on the side of the planet-sized moon that faces the planet to have only ten Earth hours of daylight during every 32 Earth hours.
There are two problems with this situation.
1) Places on the far side of the planet-sized moon, facing away from the giant planet, will have sixteen Earth hours of daylight and sixteen Earth hours of night every 32 Earth hours.
2) Places on the side of the planet-sized moon that faces toward the giant planet will have their periods of darkness brightened somewhat by the sunlight reflected from the giant planet, which should be many times the brightness of a full Moon on Earth. It is possible that might be enough to significantly reduce their hours of night.
So these are my suggestions for achieving a planet with much longer nighttime than daytime.