Let $(X_1,X_2,\ldots,X_k)$ be distributed according to a multinomial distribution with parameters $(n;p_1,p_2,\ldots, p_k),$ i.e.
$$P(X_1=n_1,\ldots,X_k=n_k) = {n\choose n_1,n_2,\ldots,n_k} p_1^{n_1}\ldots p_k^{n_k}~,$$
for $n_i\geq 0, \sum_{i=1}^k n_i = n.$
The multivariate central limit theorem tells us that
$$\frac{1}{\sqrt{n}}(X_1-np_1,\ldots,X_k-np_k)\stackrel{d}{\to} (Z_1,\ldots Z_k)\sim \mathcal{N}(0,\Sigma),$$
where $\Sigma_{ij} = \begin{cases} p_i(1-p_i) & i=j \\ -p_ip_j & i\neq j\end{cases}.$
Uniform integrability tells us that
$$\frac{1}{\sqrt{n}}\mathbb{E}\left[\left(\sum_{i=1}^k (X_i-np_i)^2\right)^{\frac 12}\right]\to \mathbb{E}\left[\left(\sum_{i=1}^k Z_i^2\right)^{\frac 12}\right]~.$$
I am interested in uniform convergence above, i.e. I want an upper bound $f(n,k)$ on the difference
$$\left|\frac{1}{\sqrt{n}}\mathbb{E}\left[\left(\sum_{i=1}^k (X_i-np_i)^2\right)^{\frac 12}\right]-\mathbb{E}\left[\left(\sum_{i=1}^k Z_i^2\right)^{\frac 12}\right]\right|$$
that depends only on $n$ and $k$ but not on $(p_1,\ldots,p_k)$ such that $f(n,k)\to 0$ as $n\to\infty$ for every fixed $k.$