Is an atomic clock itself affected by gravity? Sometimes I read that only time flows at different rates in different conditions when atomic clocks shows a different time compared to atomic clocks at altitude. But sometimes I read that an atomic clock ticks slower when gravity is higher.
Now the question is whether the number of cycles per second of a cesium atomic clock also slows down at higher gravity or that only the second is (also?) longer with higher gravity so that there is no difference?
 A: Yes, gravity affects atomics clocks.
The different rates at which time passes in gravitational fields of different strengths was tested in Chou (2010) by the National Institute of Standards and Technology (NIST) by Boulder, Colorado investigators, as reported in the journal Science. The prediction of general relativity is called gravitational time dilation (and per the link has also been confirmed by other observations).
The time dilation effect $T_d$ is approximately as follows for a given gravitational acceleration g at the bottom of the gravity well and a given height, h difference between the two clocks and the speed of light c, per the second link above:

[W]hen g is nearly constant and gh
is much smaller than $c^2$, the linear "weak
field" approximation $T_d=1+\frac{gh}{c^2}$
can also be used.

But, when clocks are in free fall, their relative rates due to gravitational effects are the same as directly measured by the NIST in Boulder, Colorado, and at 14 other locations, as described in a journal article published in Nature Physics, Ashby (June 2018).

Now the question is whether the number of cycles per second of a
cesium atomic clock also slows down at higher gravity or that only the
second is (also?) longer with higher gravity so that there is no
difference?

As another answer by @Kosm notes it is definitionally true that the number of a cesium atomic clock that passes is the definition of the second. A second is defined as 9,192,631,770 cycles of radiation of caesium-133. So, the length of the second as measured by the number of cycles of a cesium-133 atomic clock does not change.
Less trivially, the flow of time is slower in a stronger gravitational field from the perspective of an observer in a weaker gravitational field.
So, it is more accurate to say that the second is "longer" in higher gravity relative to the second in lower gravity, than to say that the number of cycles per second in the eyes of the local observer changes.
But, because the flow of gravity affects the flow of time in the vicinity of the atomic clock, this is an effect on the atomic clock. The number of cycles per second a a cesium atomic clock in the reference frame of each atomic clock, regardless of the local strength of gravity, is the same.
This is not a "mechanical" effect, it is simply a function of the fact that in Special Relativity and also in General Relativity, time passes for different observers at different rates relative to either other, based upon their speed relative to the speed of light and the strength of the gravitational field in which they are located.
While our common intuition is that we live in Euclidian space at which time passes at the same rate for all observers even relative to each other, we instead live in Minkowski space, in which time flows at different rates for different observers.
For further reading, a closely related set of answers discussion this more from the perspective of general relativity theory, can be found at this Physics.SE post.
The follow up question by the person asking the question in the comments:

But hów does gravity bend spacetime? I think GR explains according to
what mathematical law spacetimes get bended but not how this occur,
and that is the question. – Marijn Apr 25 '17 at 15:22

is basically a category error.
This occurs because it is a physical law of Nature, expressed mathematically through the Theory of General Relativity (which, even if it is not exactly correct, is still an extremely close approximation of the truth) and that is how the universe works.
If there is a mechanism that gives rise to a gravitational field (e.g. gravitons which are used to generalize general relativity into quantum gravity; although even quantum gravity assumes Special Relativity as a physical law, rather than providing a mechanism for it), this mechanism has not been observed. There is no evidence nor any widely accepted theoretical explanation that supports any "mechanism" by which time dilation happens other than the existence of a strongly empirically supported physical law of Nature which is observed.
A: Second is defined (!) as 9,192,631,770 cycles of radiation of caesium-133. But even if you define your time units some other way, flow of time in atomic clocks' reference frame is affected by gravity, that's why it would tick slower or faster compared to external observer.
A: The number of cycles per second will stay the same, but 2 clocks relative to each other will move at different rates due to the length of the second. A second that is measured in a higher gravity will take longer to an observer in a region of lower gravity. The difference is in the comparison. You cannot compare it by measuring the cycles unless you are measuring them remotely with something that would allow you to observe the cesium from your lower gravity frame of reference. As far as I know this cannot be done in real time, it is only observed when two clocks that were synchronized under the same gravity are then separated and one moved to lighter gravity and when they are returned together the difference can be observed.
Ultimately it is not the cycles per second that slows down, but instead it is the second that slows down with the cycling of the cesium.
Drew K
A: Everything in the physical world consists of a wave or a particle and therefore everything can be measured or observed, except time.  Time itself cannot be measured or observed, only the movement of particles or waves give us a concept of time changing.  We have developed different methods to measure what we call time.
   It has been proven that two atomic clocks separated by a mere 12 vertical inches show a difference in time because of the effect of gravity proving Einsteins theory, therefore, there must be a gradient of vertical time difference where the lower height is running at a slower time and the upper height is at a faster time.  This vertical time gradient would therefore exist for all bodies producing a gravitational field and as a gradient, it would be uniform.
   A single Helium atom with two stable electrons dropping between the vertical time gradient would theoretically, at one point in time, have one electron at a higher elevation and therefore in one time, and the lower electron in another slower time, and the Helium atom therefore could not exist because the two electrons would not exist at the same exact time.
   To be even smaller, the single electron would not even exist because the upper region of the electron entity would be in a different time than the lower region and it could not exist as a single unit at the same time.
Every time someone says that time is faster or slower under certain conditions, they are comparing to a standard uniform time line that is universal and never changing.  
