Time is neither matter nor energy then why it is influenced by Gravity?
Because the speed of light is constant.
I didn't get your term.
The only way light could lose energy climbing out of a gravitational field is to redshift. It can't slow down. But we can use the frequency of light as a clock (that's how atomic clocks work), so a redshift implies time dilation.
In General Relativity, gravity is geometry. It's the geometry of space-time. So, since gravity affects the geometry of space-time, it shouldn't be surprising that gravity influences time, which is one part of space-time.
This is a short and direct answer to your question, but it probably only leads to further questions. I was thinking about trying to explore some of these other questions , but the post started to get too long - so many things needed to be explained to explain more things still, and in the process the focus on the original question gets obscured.
I can perhaps list some of the other questions that might arise, even if it would be too much of a digression to try and answer them all. These would be questions like "what does it mean that space-time is unified, and how is it different from the classical concepts of space and time as being independent". And questions like "what to we mean by geometry, exactly". That one I'll give a partial answer to. Plane Euclidean geometry can be thought of in many ways, but one of the ways that's useful in this context is that it can be regarded as the study of distances. But space-time geometry is more abstract than the study of distances, it's the study of something that plays the same role in the theory that distances play in plane Euclidean geometry, but this "something", called the Lorentz interval, is not distance.
Another important question is "what is it about the geometry of space-time that makes it appear the same to us as the concept we call gravity in Newtonian physics". This is also a very good question, and a short answer might be "curvature", but I don't see how one can possibly be satisfied with this answer unless one knows what is meant by the geometry of space-time, which requires knowing what space-time is, and also knowing what geometry is, or rather specifically knowing what we mean by "the geometry of space-time". And of course there is still the question of what we mean by curvature, exactly, and after we answer that we need to explain how this curvature can possibly have effects which mimic what we used to call gravity in Newtonian theory, when gravity was just a force.
Unfortunately, in trying to answer one question, all these other questions pop up. I can't possibly answer all of them, or even try to answer them, in one coherent post. But I can raise the questions, and hope that knowing in some general sense what you'll need to learn to understand the answer to your original question is more useful than frustrating.
And to go back to the original question, since we made so many digressions - gravity affects time, because in General relativity, gravity is the geometry of space-time (and not just a force, as it used to be in Newtonian gravity). And time is a part of space-time, hence gravity affects time. I suppose I can also point out that we've done experiments (like the Harvard tower experiment) that confirms that gravity does affect time.
In my opinion your question needs some explanation. I could just as easily ask why energy is influenced when it's neither time nor matter. Or why matter is influenced when it's neither energy nor time.
I think your question is predicated on the established observation that matter is affected by gravity. But it's also an established observation that time is affected by gravity. So a better question might be why matter and time are both affected by gravity. And I think the answer might be that that's simply the nature of what we've chosen to call gravity.
Right and clear.
Matter or energy generates Gravity which is a kind of deviation from flatness of space and time.
Separate names with a comma.