Is it correct or incorrect to say that an input say $C$ causes an average run-time of an algorithm?
The average-case running time of an algorithm with respect to some distribution $D$ is, by definition, the expected running time of the algorithm when run on an input sampled according to $D$.
This should be contrasted with worst-case running time, which is the maximum running time on any input of a given length, and best-case running time, which is the minimum running time on any input of given length.
Since worst-case and best-case running time are defined as maximum and minimum, there are inputs attaining them. Average-case running time is an expectation, and so it is meaningless to talk about an input attaining them.
If you throw a die, the expected number you get is 3.5. This is not attained by any particular throw. If the die had 5 sides, the expected number is 3, which does correspond to some throw, but that doesn't mean that the throw is &quot;average case&quot;.

What sometimes happens is that you can isolate a class of inputs $X$ such that when run on an input from $X$, the running time of the algorithm is within a constant factor of the average-case running time (for this to make sense, $X$ should actually corresponds to a sequence $X_n$ of inputs of every length, or at least infinitely many length). You can say that $X$  &quot;achieves&quot; the expected running time of the algorithm.