What is the use of the Welch lower bound when k > 1?
Given any collection of $m$ unit-norm (or unit-Euclidean-length)
vectors in a real or complex $n$-dimensional vector space $\mathbb R^n$ or
$\mathbb C^n$, let $c_{\max}$ be the maximum magnitude of the
inner product between two vectors in the set. How small can $c_{\max}$ be?
One answer to this question is given by the
Welch bounds which are a collection of lower bounds on the minimum possible
value of $c_{\max}$. These bounds apply to all collections of $m$ unit-norm
vectors in $n$-dimensional space, 
and they even hold for the case when the collection is a multiset (and thus 
two vectors are identical, and so $c_{\max}$ necessarily equals $1$).

The Welch bounds are of the form
$$c_{\max}^2 \geq \sqrt[k]{C_k(m,n)}, ~k = 1, 2, 3, \ldots \tag{1}$$
or more correctly
$$c_{\max}^2 \geq \sqrt[k]{\max\{C_k(m,n), 0\}},~ k = 1, 2, 3, \ldots \tag{2}$$
because the function $C_k(m,n)$ can have negative value. Note for
example, that for $k = 1$, $(1)$ becomes
$$c_{\max}^2 \geq \frac{m-n}{n(m-1)}$$
where the right side is negative when $m &lt; n$ and $0$ when $m = n$. 
This is as it should be because we know that in a $n$-dimensional
space, we can always find $m \leq n$ orthonormal vectors and so
$c_\max^2 = 0$ for this set.
Thus, $(2)$ which becomes
$$c_{\max}^2 \geq \max\left\{\frac{m-n}{n(m-1)}, 0\right\}\tag{3}$$
is a much better way of expressing the bound.
When $m \gg n$ and so $(m-n)/(m-1)$ is approximately $1$, then 
(as Welch pointed out as also have you in your question) 
the right side of $(3)$ approaches $n^{-1}$ as $m \to \infty$ and so
we have that
$$c_\max  \geq \frac{1}{\sqrt{n}}\tag{4}$$
as a limiting form of the lower bound.

Turning to the question asked about Welch bounds for $k \geq 2$,
these are useful when $m \gg n$ and give larger lower bounds than
the right side of $(4)$. If I remember correctly, the Welch bound
for $k = 2$ gives something like
$$c_\max \geq \frac{2^{1/4}}{\sqrt{n}}$$
when $m \geq \frac{n^3+n^2}{2}$ which is an improvement on $(4)$. 

For a discussion of some of these points (as applied to
crosscorrelations and autocorrelations instead of inner products), 
I refer you
to my paper "Bounds on crosscorrelation and autocorrelation of sequences"
in IEEE Transactions on Information Theory, November 1979 which is
behind IEEE's paywall unless you have institutional access through your
employer or have paid personally for IEEExplore access.