Does there exist an approximate functional equation for the $k^{\textrm{th}}$ derivative of the Riemann zeta function, squared? That is, $\zeta^{(k)}(s)^2$.

From p.4 of (https://arxiv.org/pdf/math/0310382) a result is given for $k=1$, namely $$\zeta'(s)^2=\sum_{n=1}^x\frac{A(n)}{n^s}+\chi^2(s)\sum_{n=1}^x\frac{B(n)}{n^{1-s}}+O(\log^3x),$$ where $x=T/2\pi$ relates to the height of the zeros and $A(n), B(n)$ are certain arithmetic functions. The author states that this arises from Lemma 3 on p.29 of (https://aimath.org/~kaur/publications/15.pdf).

I don't see how the result for $k=1$ follows from that, nor how to generalise this to higher derivatives. I would expect that we can write $\zeta^{(k)}(s)^2$ in a similar form to the case $k=1$ with two main terms and an error term, but it is unclear how to do this. Can anyone explicitly give such an approximate functional equation?