The carbon cycle describes the balance of many
different geologic and biologic processes that
move carbon between the atmosphere, the oceans,
the earth, and living things.
I think your question is probably about just
one of these many processes:
photosynthesis, which is the process of plants
and algae taking carbon (in the form of carbon
dioxide) out of the atmosphere and into the
biosphere (living organisms). This process is
enabled by the harvesting of energy from sunlight.
Photosynthesis happens primarily during the
day and also primarily during the summer with
the largest number of daylight hours. So this
process is cyclical, but it is part of an overall
flow of carbon between spheres.
So, it's not that the carbon cycle works
differently at night, but the cyclical nature of
some processes like photosynthesis are part of the
overall balance of the carbon cycle. It also helps
to remember that while it's nighttime in one part
of the Earth, it's daytime in another. The carbon
cycle is on a big picture scale of the entire
Earth and atmosphere.